There's a very obvious and suspicious pause as Korra's caught between giving her name like she would normally do and maybe trying to avoid it. Though no one has recognized her (the hair cut had been a very good idea) yet, her name isn't exactly a common one. Saying it now would be equivalent to giving up the game.

In the end, her honesty wins out, and she says, carefully, "Korra." She isn't quite able to hide the quick dart of her eyes to the side, like she's checking for any reactions.