If the character has a name, I always use that, otherwise, if it's a game with a rich background/history I really want to get immersed in and all that, I try to pick a lore-friendly name. Like in Elder Scrolls games, I'll follow the naming conventions of whatever race I'm playing as. A name that fits in their world so people aren't calling me "Karen the Dragonborn", "Bob the Harbinger" or something. :P
And then in games with no obvious naming conventions in their world and no default character name, like Pokemon or Animal Crossing, I'll just use my own name or Juno. If it's a boy character I'll pick a generic male name.