It's been common practice for many years now for games to allow the player to name his/her character, especially in RPGs. Apart from Mass Effect and Zelda (I don't see the point in using any name other than Shepard and Link in those specific cases), I usually go for the same name.
I try to go for a meaningful name with an ancient Greek mythology undertone. Something with "depth" that says something, but at the same time remains appropriate regardless of what happens to me in the RPG.
I use to go by "Sisyphus". I seemed appropriately "doomed" in the case of Fallout 2, where I first used it.
However, I decided that it was too negative in connotation. After all, things didn't end well for Sisyphus (they didn't end at all, as a matter of fact).
I tried "Lovesfool", but it is too Shakespearian and it really doesn't suit most RPGs, since a doomed love is rarely the main theme, so I dropped in almost immediately.
These days, it's always "Phoenix". It is still a name with great depth, still based in ancient Greek mythology, but much more positive. It is suitable for Warrior or Mage class (maybe not so much for rogues, but you can't win them all). It is also a pun regarding loading from a previous save after each death...
What name do you use?