I keep the default name if there is one, and I'll even go to alternate sources to find a "canon" name. (For example, my Persona 3 character is always Minato/ko Arisato, after the manga. And I was disappointed that his name was changed for the movies. :() If that's not possible, I try to think of a name that fits the character.
If the character is customizable, like in Mass Effect or TES, I'll name them whatever my "favorite" name is at the time. Over the years, I've had many names that I defaulted to when I needed to name something, though only ever one at a time. Right now, it's Kestrel. I originally came up with it while playing Baldur's Gate, since the one I was using before really, really didn't fit the PC and I felt that Kestrel did, but I've discovered I actually really, really like that name. (I like the bird too.)
But if it's a male character or if there are more than one, I have to get creative. For example, in Strange Journey, my MC is Lucien Frost. And in games like Etrian Odyssey or Disgaea (for the generic characters anyway), I usually just toss together some syllables until I get something I like, or just give them random names I like. I've used Kieran in both of those, I think, and I've got a Mina in Disgaea.
And I name my Pokémon characters whichever version of their names I like best: FRLG is Red, HGSS is Kotone, B/W is either Touko or White, etc.