Character names in games. Do you name them and does it matter?

[Arcrom]

Personally I don’t bother much in giving custom names for characters if they already have a predefined one. I think the only exception was with Fire Emblem Rekka no Ken where it simulates that you are there as a strategist that witness and lives the events of the game with people looking and talking with you (all this with first person view). This gave me some reason to put my own name for him instead of the default one.