In Habitable Planets for Man, Stephen Dole establishes that a satellite planet in orbit around a massive gas giant, (like Jupiter's Titan, I guess) would be habitable for humans.
What would it be the effect of the main planet on the satellite's climate?
Is it mandatory for the satellite planet to be tidally locked?
Short answer
The tides would be enormous, so you'd probably need tidal locking and that would likely lead to longer days and the effect of longer days, lets say, 40-100 hour day/night periods vs 12 hours on Earth might be problematic and could effect weather and plant life, and lead to higher daytime temps and lower nighttime temps, so more frost and more heat death. Those are the biggest issues, either 100 foot tides or week long days. Pick your poison cause it's one or the other.
There would be other hiccups, like a deeper gravity well, so faster meteor impacts on average and a more chaotic system 3 body system, so more stirring up of "Near Earth objects", perhaps leading to an increase in not only severity but frequency of large asteroid/comet impacts. But generally, I think life on a habitable moon orbiting a gas giant would be possible. Perhaps not ideal, but possible.
Habitable worlds.
"Habitable" is kind of a big word and some definitions should be made. A habitable moon, suitable for man, as you state in your question, implies a relatively dynamic object. Too light and it's likely to lose it's atmosphere, and humans get disoriented at less that 1/6th G force. Our inner ear/balance system gets thrown off.
At the very least, a habitable world should have a breathable atmosphere, it probably should have an ozone layer, though protective clothing and eye-wear could be worn as an alternative, and it probably should have oceans, though a desert world could be considered habitable through water extraction from underground, from polar ice caps or from the atmosphere.
A magnetic field is a big plus, and probably serves, at the very least, to help protect the Ozone layer and perhaps the entire atmosphere over time. Plate tectonics, while arguably not essential to a habitable planet, it's thought to be necessary on a planet where advanced life develops. It's useful for recycling molecules absorbed by life that settles on the bottom of oceans.
Finally, a habitable world needs a proper rock to water ratio. Too much water and you have a water world, which could be lived on if you don't mind living in boats, but I wouldn't consider a water world as a habitable world. Feel free to disagree if you like. But too little water and you have a planet that's a desert, too much, it's just one ocean all the way around. The proper water to rock ratio is surprisingly small.
like Jupiter's Titan, I guess
Titan or moons like Titan don't work. Titan has a solid surface because it's very very cold, but if you moved Titan into the habitable zone, all it's frozen Methane, Butane, Ammonia and CO2 would thaw and a lot of it's water-ice would melt and Titan would become a small ocean world with a crushing atmosphere. It would also lose it's atmosphere fairly quickly in the habitable zone, at least on a solar-system time scale. It's not 100% clear why Titan has an Atmosphere, but Jupiter's 4 large moons essentially don't, but Titan's it's temperature (about 93.7 K, is comfortably above the 77 degree K boiling point of Nitrogen, as opposed to Pluto or Triton where Nitrogen mostly remains frozen, but Titan may have been cold enough throughout it's history to retain an atmosphere. It might have also, been a fair bit larger when it formed and it may have lost a considerable share of it's atmosphere over time, but I digress. Atmospheric loss is relevant to your question, but it's a complicated subject.
Rocky worlds vs solid-ice worlds.
There's a distinct difference between rocky bodies, like the 4 inner planets and Earth's moon, and the all the large moons around gas giants in our solar system and known dwarf planets. The large moons other than Earth's moon and all the known dwarf planets have vast amounts of water. Many have underground oceans.
The water to rock ratio is much too high for moons like Ganymede or Titan. A habitable moon/planet should probably have a water to rock ratio somewhat similar to Earth.
Ganymede has a solid surface, but that's because it's lost any surface water/ice due to exposure to sunlight and fast moving particles in Jupiter's magnetic field. Not for below it's dry surface, Ganymede's outer layer is basically frozen dirty ice and below that, a liquid salt water ocean and below that, funky solid hot ice. Like Titan, Ganymede would be a ball of dirty/salt water with a thick atmosphere if it was moved into the habitable zone.
A habitable world would likely need to have either rocky body formation like the 4 inner planets or it would need to have lost a vast amount of it's ice by some means after formation perhaps by migration too close to the sun, then back out again. A proper rock to water ratio is pretty much non negotiable for a habitable planet, unless you want to live in boats on the surface of a water world and extract all needed minerals from the ocean water. Somewhere in the range of 1000 parts rock to 1 part water seems about right for a habitable world. It's unlikely that 1000 to 1 ratio would form outside the frost-line, and it's thought that most gas giants do form outside the frost-line, so one possible scenario of a habitable moon around a gas giant planet would be capture.
This has nothing to do with your question, but I mention it only to point out that rocky/habitable worlds around gas giants are probably rare. Capture is possible, perhaps some kind of 3 body dance, could happen during planetary migration. Rare, but not impossible.
Longer Answer and Factors
In a nutshell, the question could be summarized as, could Earth remain habitable if it orbited Jupiter and what would be different? I'll use Earth, Jupiter and our Sun as examples to avoid additional variables.
Stable orbital distance, aka, True Region of Stability and Tides
If we use Jupiter and Earth as examples, Jupiter is about 318 times the mass of Earth, and we know Earth's true Region of stability is 1/3 to 1/2 of it's Hill radius, or about 500,000 to 750,000 km.
The Hill Sphere radius increases with the cube root of the mass, so 318 times more massive (Jupiter to Earth ratio) works out to about 6.8 times further out. That puts an outer limit on the stable orbital distance of a moon around Jupiter at Earth's orbit around the Sun of about 3.4 to 5.1 million km. That's the outer edge, you might want to say closer than 3 million km just to be on the safe side.
At 3 million km from Jupiter, or about 8.7 times further than Earth is from it's Moon, and Jupiter's mass being over 25,000 times the mass of the Moon, the tidal force Jupiter would exert on Earth would be 40 times currently exerted by the Moon. The effect of 40 times the tidal force could be an entire question in and of itself, and that would be about the minimum that Earth would get, orbiting Jupiter at 1 AU from the sun. That would be problematic without tidal locking, so in your scenario, you'd probably want Tidal locking, which make the tides more like permanent bulges and less like moving train wrecks.
Another tidal related problem is libration. You'd probably want to move the Earth into a closer orbit to Jupiter, more like 1 maybe 1.5 million KM to lock the orbit into a more circular orbit with less solar perturbation. Moving into a closer orbit would significantly increase the tidal bulge but significantly decrease how much it moves around, and movement would be a bigger problem. In such a scenario, there would probably be a tidally driven permanent ring of fire, but Earth has plate tectonic driven rings of fire now without any real threat to life, so I don't think that, in and of itself, would be a big problem.
So, on tidal locking, I'm leaning towards a hard yes, unless you want to deal with ridiculous tides of 100 feet or more twice a day and probably some tidally driven earthquakes, or if you create a variation on the theme, a hot star, maybe twice the mass and 16 times the luminosity of our Sun, so you can move the Jupiter some 4 times further away (or 5 or 6 times with more greenhouse gas), then you can move your habitable moon at a more distant orbit from it's gas giant and avoid the need for tidal locking, but it would only work with a more massive star, and that would considerably shorten the life-span of the solar-system.
But, with a sun like star, our best scenario is an Earth in a close to perfect circular orbit around Jupiter, maybe about a million km away. That's roughly the distance of Ganymede to Jupiter so the orbital period would be similar too, about 7 days.
Longer Days/Nights
A day on our theoretical Earth would be 7 days, 84 hours of sunlight, 84 hours of night on average. You could move the habitable moon closer to Jupiter if you wanted shorter days, but there's another problem that would make this scenario even more improbable. Gas giants tend to rotate very fast due to conservation of angular momentum. A rapidly rotating gas giant would have a tidal bulge that would lead the massive habitable moon, and that tidal bulge would accelerate the Moon's rotation, transferring angular momentum to the moon and causing it to move away. A moon of significant mass, and for a habitable moon, that's a requirement, this movement would probably be significant. Even at a million km, the movement away from the gas giant might be significant, at least in terms of planetary ages. At some point, the Habitable moon likely moves far enough away from the gas giant to lose its near-perfect circular orbit and tides would become a problem. That movement away would take many millions of years, but it still raises the problem of how likely this scenario is.
Granted, you might imagine a slow rotating gas giant where the two objects enter mutual tidal locking. That would probably be ideal, but slow rotating gas giants are probably rare.
Ignoring that for now, If Earth was moved into an orbit around Jupiter at 1 AU from the sun and 1 million KM from Jupiter, it would orbit Jupiter approximately every 7 days, which would mean longer, hotter days and colder nights. Perhaps increased wind speed due to greater temperature variation and a change in the Hadley cell set-up might be possible due to a decease in the Coriolis effect, perhaps changing from 3 Hadley cells per hemisphere to one like Venus or maybe two. The effects on weather would probably be considerable, but too complicated for me to guess. Maybe Earth would stop having Hurricanes as a wild guess.
Too much heat is bad for plants and so is too much frost at night, but plants would probably make some adaptive measures if they evolved on a planet such as this one. It might be more difficult if we tried to settle on a planet like that, as life on our planet isn't used to 7 day "days" but I think it would be more of a big inconvenience than a total deal breaker.
Seasons
The planet could still have 4 seasons. If Jupiter was at a 23 degree tilt and the Earth orbited Jupiter's equator, then you'd still have seasons like we have now, so winter/summer/spring/fall would be largely unchanged, but only if your gas giant's tilt was just right. Jupiter itself has an axial tilt of just 3.13 degrees. With that tilt, there would effectively be no seasons.
Magnetic field
Earth's magnetic field would largely protect it from Jupiter's magnetic field, and depending on how close, there might be permanent auroras, which might be nice, but could make star gazing more difficult. Without a magnetic field, the atmosphere would be subject to stripping, not just from the solar wind but from the gas giant's magnetosphere. A magnetic field on the habitable moon is strongly encouraged.
Night sky and Eclipses
In addition to the possibility of permament auroras, Jupiter in the sky would be enormous, some 10 times the angular diameter of the Moon in the sky and 100 times the surface area, about 5 degrees of Arc. Jupiter would probably be pretty dark up close, kind of a deep brown color at habitable zone temperatures, but the Moon is the color of dark asphalt and it glows white in the night sky. It's hard to say precisely what Jupiter's brightness would be at night, but it would probably reflect enough light on Earth to read by and plants might even develop a 2nd set of leaves or flowers for the 84 hour nights. Basically night leaves and day leaves.
Eclipses would be much more frequent and Jupiter's Umbra would be nearly the size of Jupiter at 1 million km distance, so a well lined up eclipse could last 2 hours and the people on this Earth could get eclipses that repeat one day and the next day (7 days later). At low axial tilt, like 3 degrees, perhaps less, Eclipses might happen every day, but 2 hours of eclipse from 84 hours of sunshine probably wouldn't have a big effect on climate. It might even be a benefit, happening at peak sun every day.
The Meteor Problem
Earth has an escape velocity of about 11 km/s. Earth orbits the Sun at about 30 km/s. Those two numbers and a little bit of math means that a meteor or comet striking the Earth, unless it's from another solar system, which is much more rare, the velocity range is a minimum of 11 km/s and a maximum of about 71 km/s, and most orbits in the same relative direction of the Earth's orbit, the impacts are closer to 11 km/s.
if Earth orbits Jupiter at 1 million KM, it's orbital speed around Jupiter would be about 11 km/s, which could add to or subtract from it's 30 km/s orbital velocity around the sun. Also, in addition to Earth's circular orbit around Jupiter, Jupiter's gravity well at that distance would be about 11 x 1.414 or 15.5 km per second, so for any objects coming from outside the Jupiter orbit, the minimum impact velocity would be 25.5 km/s and the maximum impact velocity, about 11 + 11 + 15.5 + 30 + 30, or 97.5 km/s. Objects already in orbit around Jupiter would be less, but for so called "Near earth objects" or NEOs, the velocity of impact goes up if Earth orbits Jupiter.
You'd also have a chaotic 3 body system. As it stands now, Near Earth Objects are mostly in stable orbits, orbiting the sun, never getting too close to the Earth. That's why we have relatively few large asteroid impacts. If you moved Jupiter into Earth's orbit and set Earth orbiting Jupiter, Jupiter's mass would much more significantly stir up the NEOs. Granted, Jupiter is good at expelling things from the Solar-system, and it would do that with NEOs too. There would be less of them, but the ones that weren't expelled would be in more chaotic orbits and, while predicting this is extremely difficult, my guess is, in such a scenario, large asteroid impacts would increase, not just in frequency but also, on average, in velocity.
Summary
Mostly these aren't deal breakers. Earth could be hit by a very large, end of civilization type of asteroid this week, but it's statistically very unlikely. An increase in the likelihood of a very rare event is still a very rare event, but asteroid tracking would be more difficult and mass extinction impacts would probably happen more often.
The potential tidal problem and week long "days" are the biggest issues I see, and the fact that a rocky world with the right amount of water might have a very hard time settling into a steady orbit around a gas giant, but other than that, I don't see too many problems and a habitable moon for man-kind is theoretically possible. A habitable ocean world, while less good for us, is probably much more likely.
There have been some questions about potentially habitable exomoons of gas giant exoplanets in the habitable zones of other stars.
The most important factor to consider is that a moon would have to be in the size range of a habitable planet in order to be habitable. And the moon and its planet would have to be orbiting in the habitable zone of their star. And there would be other factors to consider.
I have an answer to this question:
Can a gas giant have its own habitable zone?1
That links to other similar questions and to an article on the subject.
And here is a link to a less technical article:
https://www.skyandtelescope.com/astronomy-news/habitable-moons/2
