Well keeping in mind that the question is simply being the character, and not necessarily in their world, thus in the real world, then I would have several different answers given that I'm the only person in the world that's been given this opportunity for no apparent reason.
!) Alex Mercer has a sweet shape-shifting arsenal. Shape-shifting in general is awesome to me, plus all the cool stuff he already figured out he could do with it. I'd probably learn how to change into things at will, instead of having to consume a target to change.
2) Pretty much anyone from the future so that I could "invent" all their gear for profit and lulz. Advance mankind, while making me rich. I guess if I had to be specific, I'd go with Samus. Even if I couldn't have the ship with me, She takes place so far in the future that 1 manned ships can break the light barrier, therefore she must know how to fix her ship from time to time when it breaks down. (maybe that's why it changes so much) At the very least her knowledge of the tech she uses regularly and education from the future would put in a good position in the world.
3) Well, you said ANY game, and this thread is missing something... http://www.youtube.com/watch?v=CSlnZxvi37s&feature=player_embedded Freeware is still ware, and it's still a game. ALthough I'd honestly only say this if I got at least 6 other friends this opportunity to become a character. Being the only one from that game in a normal world would be awkward. Although I could still use magic to try to advance mankind, and I'd probably become really popular for varying reasons.
Overall I'd probably go with Alex, since I could probably in time learn how to shape-shift into whatever I wanted to. Hell, I could get jobs at companies for making life size or scale models of cars or buildings and such. Then go to Cons with perfect cosplays during vacations. Plus research on myself could probably enhance mankind to, as long I don't start a Zombie Apocalypse. Well at least I'd have some food then.
Oh, and virtual immortality. That's nice, and it's the good kind too. Like, I don't age, I can still feel, but I heal really quickly, and if I really wanted to die at some point, I think that there are ways.