Most sci-fi games that involve interstellar travel (other than as a one-way trip) handwave this away by counting on the willing suspension of disbelief on the part of the players, sometimes combined with the likelihood they won't know enough about relativity to know why their handwave doesn't really work. For instance, the jumpgate/wormhole/etc. approach feels neat and tidy; no one really goes faster than light, right? They just take a "shortcut". People who don't know relativity are convinced this is a way to avoid its limitations, but in fact, this doesn't quite work: specifically, like all other kinds of FTL, it is fundamentally equivalent to time travel, and therefore far outside the bounds of physics as we understand it. Nothing wrong with that, of course; it's just that most people running a wormhole-based SF game don't realize that they're one step away from time travel and the attendant complications and separation from "real world" physics.
My favorite handwave is one I came up with a few years ago. It in no way avoids the time travel paradox, but it's very simple and elegant, it leads to some interesting consequences, and I've never seen it used. Speaking mathematically, the impact of time dilation as one approaches the speed of light in vacuum is fairly similar to the part of the curve of an tangent function between 0 and /2, in how it climbs to infinity as it asymptotically approaches a vertical. (The curve isn't exactly the same as that, I think, but the asymptotic approach is the same.) Advances in physics often involve realizing that what was known before was true as a special case of a more general rule; the best example being how relativistic physics approaches classical Newtonian physics as velocities approach zero. What if relativity turns out to be a particular case of a more general rule in which the effect of time dilation turns out to resemble the whole tangent curve, with c (the speed of light in vacuum) corresponding to /2? Then dilation reaches infinite values at c, but is zero at 2c, infite at 3c, zero at 4c, etc. Assuming a drive that can skip discontinuously over the intervening velocities (i.e., infinite instantaneous accelerations, but hey, this is a handwave, right?) but at an energy cost proportional to the number of multiples of c jumped over, you can get places awfully quickly without any messy time dilation if you can spare the energy. Nice and tidy. Trouble is, how many people would understand it? I might as well call it a heebie-jeebie drive.
Assuming you want to run an interstellar sci-fi campaign within our current understanding of relativity, there is one very easy solution that hardly anyone even considers. Simply posit a species that is exactly like humans (or isn't, as you see fit), except for one thing: they have a metabolism one one-hundredth the speed, and a correspondingly longer lifespan. They live on a world which rotates and revolves that much more slowly, too. There are some side-effects of living at a hundredth the speed which would have to be considered (there are any number of physical processes, like chemical reactions, which would still happen at the same rate), but it'd be feasible to have a society and culture just like humans but at a hundredth the pace, so that a c/2 trip to Barnard's Star with nearly no time dilation to worry about would take the equivalent of 51 days. There, all solved!
However, I've always found it compelling to run a campaign which embraces time dilation as a premise. I don't mind positing FTL for these campaigns, but time dilation is still a key point. Here are the two ideas which I have played with, half-developed, but never completed and run.
The Long Road: In the near future a colony-ship is launched using technologies we could almost build right now (if we had the will and money) to travel at near-lightspeed to an eligible planet for colonization, to preserve humanity through an anticipated global disaster. While the crew is incommunicado due to time dilation, the disaster is averted, hundreds of years pass, and a method of FTL travel without time dilation is discovered. The characters emerge not to the wild frontier of an empty planet ready to be colonized, as they expected and prepared for, but a fully colonized planet, part of an interstellar civilization of humans. I've seen this idea dabbled with before, but never really explored as well as I'd've liked.
The Empty Sky: Same starting premise. However, during the hundreds of years, expanding humanity has encountered another species of interstellar travelers. Something happened since which, inexplicably (at least at first), caused every human and every one of these aliens to simply vanish instantly and simultaneously. The slowship travelers are the only exception, having been preserved from this fate by a form of dilation not known to Einstein by which complexity, and even intelligence, diminishes as one approaches the speed of light (a weird jumble of quantum mechanics, relativity, and information theory -- it made more sense when I wrote it all out). Now they find themselves travelling the empty halls of a vanished civilization gathering clues about what happened, and maybe eventually finding a way to undo it.