For my story, I'm trying to figure out where to send a probe that might have a chance at finding an alien civilization similar to ours. What is the closest star system to ours that might have a chance at supporting life?
Question being answered: "What is the closest star system to ours that might have a chance at supporting life?" the answer is Proxima Centauri
"Proxima Centauri" - https://en.wikipedia.org/wiki/Proxima_Centauri -
In 2016, the European Southern Observatory announced the discovery of Proxima b, a planet orbiting the star at a distance of roughly 0.05 AU (7,500,000 km) with an orbital period of approximately 11.2 Earth days. Its estimated mass is at least 1.3 times that of the Earth. The equilibrium temperature of Proxima b is estimated to be within the range where water could exist as liquid on its surface, thus placing it within the habitable zone of Proxima Centauri, although because Proxima Centauri is a red dwarf and a flare star, whether it could support life is disputed. Previous searches for orbiting companions had ruled out the presence of brown dwarfs and supermassive planets.
The Breakthrough Starshot initiative has plans to send a probe there - http://www.space.com/33844-proxima-b-exoplanet-interstellar-mission.html-
The founders of the Breakthrough Starshot initiative want to send wafer-thin probes to Proxima Centauri at very high speeds. The plan calls for equipping these probes with thin sails, which would capture the energy imparted by a powerful Earth-based laser.
This laser would accelerate the probes to 20 percent the speed of light (about 134.12 million mph, or 215.85 million km/h), according to the program scientists. At that rate, the probes could reach Proxima Centauri in 20 to 25 years.
An updated approach is this "Studying Proxima b: Tiny Sailing Probes Could Orbit Nearby Exoplanet" http://www.space.com/35549-proxima-centauri-mission-solar-sail.html
The original Starshot plan calls for missions to this planet (known as Proxima b), or to any other destination, to be flyby affairs; the nanoprobes would snap photos and collect other data as they hurtle by at breakneck speed. But it doesn't have to be this way, according to the new study, which was led by René Heller of the Max Planck Institute for Solar System Research in Göttingen, Germany.
Heller and co-author Michael Hippke, an IT specialist, performed computer simulations showing that Starshot-like probes could slow down enough at Alpha Centauri to be captured into orbit there. This deceleration would come courtesy of the binary's starlight pressure — which would push back on the nanoprobes' sails, just as outgoing photons would have pushed the spacecraft forward at the beginning of its trek — and the gravitational pull of the Alpha Centauri stars.
This has been proposed well before any of the new knowledge existed by Larry Niven & Jerry Pournelle in "Footfall" https://en.wikipedia.org/wiki/Footfall
Footfall is a 1985 science fiction novel by American writers Larry Niven and Jerry Pournelle. It was nominated for both the Hugo and Locus Awards in 1986, and was a No. 1 New York Times Bestseller. The book depicts the arrival of members of an alien species called the Fithp that have traveled to our solar system from Alpha Centauri in a large spacecraft driven by a Bussard ramjet. Their intent is conquest of the planet Earth.
I have seen more than a few ask about this "Bussard ramjet" well the Book gives a good description on how it works. The Book also uses "Herd mentality" which of course could open a few questions here.
I do not know the legal requirements or etiquette but ... I will inlcude this in case it spurs some creativity
They possess more advanced technology than humans, but have developed none of it on their own. In the distant past on their planet, another species was dominant. This predecessor species badly damaged the environment, rendering themselves and many other species extinct, but left behind their knowledge inscribed on large stone cubes, from which the Fithp have gained their technology. Facing possible extinction due to the long-term effects of biological weapons, a group of high-ranking Fithp were selected to escape to the stars
Your Probes Better Have Good Range
If there were a civilization as advanced as us nearby we would almost certainly know about them. Radar from earth could be picked up at a distance of 15 light years with current SETI technology and up to 250 light years with proposed systems. Other forms of highly directional broadcasts may reach even further than that.
As far as possibly supporting life look no further than the Kepler Mission, its catalog has hundreds of rocky worlds that are the goldilocks zone. These are not sure fire things, but they are the very best bets for life as we understand it.
For finding intelligent life at our own tech level in another star system: we probably won't, ever. It would be like finding a needle in a haystack made out of more haystacks. And the needles are radioactive and keep decaying into hay. And that's ignoring the time component. Ie a civilization as advanced as our own, existing at the same time as our own has a virtually zero probability of occurring.
For example, if you imagine the history of the universe compressed down to the length of your arm, if you took a nail file to your middle finger, oops, there went all life on earth. If the history of all life on earth were the length of your arm instead, the same nail file would have removed all human history. If instead all of human history were the length of your arm, the nail file would have removed everything since the industrial revolution.
We've been searching the skies for alien radio for 50-odd years now, and not a blip.
Getting a prove to another star system would take as much time as the length of your fingernail in this last comparison. Roughly. I'm not in a position to do a precise calculation at the moment, but it gives you an idea of just the shear scale of the problem.
Realistically we'd send probes out to every system with an earth sized rocky planet in the habitable zone nearer than 100 light years. We MIGHT get lucky and find something. For your story, we do. Problem solved. The thing is, they wouldn't be aimed, they'd be scatter shot. Unless they were von neuman in design (see: We are Legion (We are Bob).)
I would say your best chance of a nearby life-sustaining planet would be a planet whose life has not explored the communication options presented by radio - it is nearby, but simply hasn't been detected with conventional means.
There are two main candidates that are relatively well-known - Proxima Centauri's planet Proxima b (described in detail by other comments), and the furthest out planet of the newly discovered TRAPPIST-1 system (39 light years away).
The reason that despite being proclaimed as 'having several planets in the habitable zone' only the furthest out is habitable is that the solar wind has the impact of moving the habitable zone further out.
TRAPPIST-1h (the furthest out planet) orbits its parent star every 18.7 days, it orbits at an average of 0.06 AU, its radius is 0.76 times that of earth and its mass is unknown. It is most likely a rocky planet as no traces of hydrogen have been found in its atmosphere.
One of the other pros of the TRAPPIST-1 system is that the other planets are easily visible in the sky ('No Man's Sky proportions') and are only kept from destabilisation by orbital resonance.
Because of this, a physical probe like one in the starshot/starwisp program has a good chance of catching several of them in one photo by simply aiming for the star and expecting to miss slightly.
All that would be needed to confirm life would be a photo showing clear evidence - an artificial structure like a large dam, or (from a photo that can be further away) flora with a clear identifiable colour.