My own best shot is for humans to first make themselves immortal by genetic manipulation (which is almost certainly impossible to accomplish in the real world but this is a game) and then create a sort of generation ship that can carry the smallest number of people necessary to preserve genetic diversity - about 200 individuals - to a nearby star. The ship would need the most powerful energy source one can conceive as technologically feasible, nuclear I suppose, with a huge reservoir of uranium or perhaps thorium. I suspect that crunching the numbers will show this to be unrealistic, but it should be close enough to science to pass muster.
You could try a true generation ship, with hundreds or thousands of generations of people being born and dying during the trip, but somehow I can't see that working out from the perspective of social stability. You just need one bout of anarchy or one lunatic to sabotage the ship's drive and it's all over. Immortals have a better chance, since time has this habit of speeding up the older you get, and if you've adjusted to the first few decades of travelling in an interstellar ship then you're good for the next several millennia.
BTW the responses on the thread are interesting. This pessimism didn't exist a few years ago.