The species doesn't originate from the solar system, nor do they occupy a single planet or even have any colonies on the planets. They travel around in space ships throughout the solar system
Do they just choose an arbitrary planet and use the rotation and orbit of that planet to count the days/years? Do they use their home planet's date/time (even if it no longer exists)? Use an arbitrary date/time system based? Are there existing date/time systems that are not planet specific?
Since there is no absolute starting point for time, nor any absolute standard unit for it, you would definitely need some reference point for when you're counting from and how long each unit of time is.
All species would have to have originated from somewhere. This is most likely to be a planet of course, but it can be a space construct in fantasy setting. If it is a planet, then that planet will have a day-night cycle, or in the absence of such a cycle, will have a yearly cycle. Having the amount of time in a year as a starting point and having to have subdivisions to count smaller intervals, they would already have a system of timekeeping.
Without any reason to change it (since time intervals don't change), I assume any civilization that leaves its homeworld would just use the same date/time system. It's familiar to them, computers and mathematical systems would already work with it - it's one of those "if it ain't broke, don't fix it" cases.
However, it may be problematic in some cases. For example, you might need a way to compensate for clock drift or be able to determine not only where you are but when you are, if there is a chance your systems are drifting or you need to identify how much time has passed from an event. There would be 2 ways to accomplish this, one being dead reckoning based on what position you think you're in (you check the stars around you, compensate for your movement and see how they've moved, which, assuming your theories about orbital and stellar mechanics are correct, could give you a length of time) and the other would be to have multiple accurate clocks and compare them. Both would probably be used - but is that practical with a date/time system used on a planet?
Probably, since we can just have a format for time and date used for calculations and then change it into a more easily used format for speech (this is what computers do). A long-time space-faring civilization might come up with alternatives, to get over the need to perform complex calculations or constantly compare with their own planet.
A simple change would be to determine what the shortest day is for any planet in the solar system (for planets that are not tidally locked, because those lack a day-night cycle) and use that as the base time unit for days (It might be preferable to not call them days in this case of course). This can be taken further, changing the year length to the amount of time it takes for the closest or farthest terrestrial planet to circle the sun, or the one with the most circular orbit. The intervals for our minutes and seconds could also change - many time units have already been expressed as multiples of Planck time and the same could be done here, albeit with some larger interval instead of the theorized smallest amount of time in the universe.
Of course, this approach would mean that as time goes on and more planets and solar systems are discovered, these times would likely be changed to some other average. Perhaps this species can predict some average times for years, days etc. and create a time/date system that is practical, being easy to use on most worlds.
Division of time into days etc. is a convention. Other species need not follow the same convention. Indeed, a space-faring race is going to have to come to grips with a more perplexing reality than how to break up time: how to even talk about time given relativity.
The only real solution is to maintain a local clock that is synced with whatever important biological (or electronic?) processes occur within their bodies, so that "one year" locally means approximately the same thing in terms of growth, aging, whatever, regardless of the location; and that everyone know how to compute arrival times in the local clocks of anyone they want to interact with. So, for instance, if these were humans, we'd probably keep Earth Time, but not use it around, say, Betelgeuse.
As far as what units to use: our units are chosen out of convenience in part. We need to sleep every ~24 hours or so, whether or not there is a perceptible day, so it is very convenient to have such a unit. We also can do quick things an about one second, so it's good to have something about that long. Years are dispensable; they matter a lot when on earth, but not elsewhere.
So humans would be well-served by a time system based on days. You could use 10-5 days to replace seconds (100,000 centimillidays per day vs. 86,400 seconds per day), and not worry about anything else. If you're in a local situation where some other period matters (local day or year or whatnot), just use both like we do with days of the week.
Aliens might not count in base 10 (it's actually rather awkward for division of small numbers), and if sufficiently advanced there may not be any periods left that matter any more, but if they are, those would be best to use.
I am surprised no one has answered with nuclear decay as a basis for time keeping. For a nomadic space faring society the local orbital/rotational times are going to be rather unimportant. They are primarily going to be concerned with their biological requirements, shipboard routines, and keeping rendezvous.
I would think that picking an isotope with a half-life convenient to the biological cycles of the species as the bases of your time unit makes a lot more sense than trying to adjust to local solar cycles every time you change stars.
About the only cyclic event that seems a likely measure in the described scenario is the previously mentioned Galactic Year.
Edit: It occurs to me that I am assuming interstellar travel, which the OP did not specify. I still feel that decay would be a viable option for a society that is bound to a single star, but another option that has not been expressed would be planetary alignments. Using the solar system as a whole for your clock with the times when certain planets align being the significant dates such as "New Year".
Are there existing date/time systems that are not planet specific?
This part of the question doesn't seem to have been answered yet. Yes. They're all counts of units of time since an arbitrary point (an epoch).
Julian Days are the number of days since November 24, 4714 BC. Similarly, Rata Die is days since Jan 1, 1 AD.
Getting away from solar days, Unix Time is the number of seconds since midnight Jan 1, 1970.
Astronomers use a Julian year which is defined as 86,400 SI seconds but it's not meant as a calendar unit, merely as a fixed unit of time which does not vary.
Do they use their home planet's date/time (even if it no longer exists)?
Ultimately, you have to pick some arbitrary unit of time as the basic unit of your system. We have chosen the second and then defined it based on various physical properties. A space-fairing civilization would likely have similar historical baggage.
If the species has a Circadian rhythm like we do, they would likely retain the day cycle of their home planet to continue to match their work/rest cycle. Otherwise, there's little reason to retain it.
There being no seasons in space, years and months would have less relevance, but there would need to be a way to easily communicate large chunks of time. Earth has annual seasonal cycles, and monthly lunar cycles. These cycles were and remain very important to our society, so we would retain them in our calendar as we go to space. A species from a planet with no such cycles, or ones of less significance, would not bother with them. They may come up with an arbitrary way to express large chunks of time.
Humans use calendar cycles for scheduling and regulation of work periods. Another species may not do that with cycles and thus would have no need of a calendar with cycles in it.
Any civilization capable of living on multiple planetary bodies is capable of the calculations required to convert between different local times. One option is to have an atomic clock on a satellite in a stable orbit broadcasting a single point of reference - timestamps using the satellite benchmark keep a universal point of reference for everyone in the system. Yes, it could be on a planet, but being its own satellite removes that favoritism.
Clocks would be tracking whatever time period is most meaningful for the locals. This may be largely based upon their planet of origin, simply due to biological necessities and tradition, but should be whatever works independently for that habitat. Local time is the only time really relevant to people, and it is trivial to convert this benchmark to be expressed in local time when data is transmitted/received.
One amusing factor would be the apparent inconsistency of the benchmark signal - as a ship moves around the system at various speeds or landing on different planets, the universal clock might seem to be slightly faster or slower instead of keeping stable tick, but this would be a tiny variation and easily anticipated for automatic adjustment when converting between different 'local' times by a computer systems.
Are you basing your time on motion or light? The moon's cycles and the zodiac months? At the point the difference between motion and light become important enough to make an adjustment comes into play will only be important if they make use of information from far enough away.
Knowing at what speed the whole history of what is currently perceived is on its way to us in our future when making use of light or any other spectrum's recorded speed, we use the largest currently understood and confirmed notation to unite actions.
I think I read this question more as "What shall we base time on now?"
"One amusing factor would be the apparent inconsistency of the benchmark signal"-pluckedkiwi
"Complete simultaneously is not possible, with differences of up to half a microsecond." - JDglugosz
Time has always been only a reference point to something. The units of difference being related to the most consistent changes so any race's timescale would entirely depend on their mutual acknowledgement of some perception's consistent difference and its accuracy. This can be Anything really. Just think about the range of timescales in musical notations and compare them to the length of the song ;)
Enjoy
Do they use their home planet's date/time (even if it no longer exists)? Use an arbitrary date/time system based? Are there existing date/time systems that are not planet specific?
Lets see the problem from practical position. If I would be the space traveller without home. And possibly humanoid or about it. I need time units for: sleep-wake cycle, ship maintenance cycle and lifetime cycle.
Sleep-wake cycle is clear. Choose any value which is good for your race. For more information see Michel Siffre studies. In his experiments he live in cave for a six months and in result he has very different sleep-wake cycle.
Ship maintenance cycle - this is time for measuring ship fueling time or space travelling time. It is useful to measure it in tens-hundreds-thousands of sleep-wake cycles. On Earth we have moon cycle. It is not super precise in long-term but in ancient civilizations it was useful. As I remember on Easter Island time was measured in "moons".
Lifetime cycle - you can choose any big cycle events which you can see in space. It can be pulsar clock. From wikipedia: "However, for some millisecond pulsars, the regularity of pulsation is even more precise than an atomic clock". Or you can see binary star pulsation. Also pulsars and binary stars are not planet specific time measuring.
