Companions

So, at what point do your game avatars become characters?

Is it when the game's story evolves them to the level that they finally have enough depth?  Is there a certain hour-mark where they become worthy of things like names?

What makes a character, from a player's point of view?  I have friends that spend a lot of time on first-person shooters that have a favored weapon loadout and may spend time customizing their appearance, but I can't think of any time that they've referred to their avatars as anything as familiar as "my character."

When do they transition from coded construct to personality?