In a single-player game with no online interactions, I don't care what others name their characters (unless they're exceptionally funny, brilliant, or stupid).
I'd like a first and last name slot, because, for example, if I enter a first and last name in one box and a character who's supposed to be, say, my character's love interest, calls me "Jared Tinson" rather than just "Jared," I might feel a little awkward. Unless it's polite for lovers to address each other by both names in the Project Eternity universe. Pretty much what Kabaliero said.
Also, a name generator would be cool if I want a name that fits with the game's lore (more than my own name or Jared Tinson or something) and don't feel like digging through the lore to figure it out myself.