If it is more story than game, I like having a name; if it is more game than story, no name is needed.
Example:
The Wal-Mart Game does not name the main character; the main character is you. Anyone can visit Wal-Mart.
The main character of The Price of Freedom: Innocence Lost is named; chances are the player is not, nor ever will be a gladiator.
There are exceptions to this that work out quite well.
In Eternal, the protaganist's name is never mentioned (within the game), yet because of the way it is written and the context of the story, it never seems unnatural.