First playthrough, I make something up or use a cool name from an anime, book, or another game.
After a few runs, I go for something funny that goes with the text (e.g. "my dude, ").
For arcade games
High score/good personal score: DAN from the name I had to come up with in Chinese class years ago.
Decent score/just missed the high score: DAN or CFY, "Comin for ya". My brother came up with it, to make the record holder quake in their boots knowing that their status is in jeopardy but because of the limited characters you wouldn't know that so it's really my own personal joke.
Embarrassing score: sometimes I own up to it, sometime it's AAA.