One way you can go with this, and this is completely off the cuff, is to turn it back on her. Kinda like the martial arts principle where when you're attacked, instead of backing up, you move in closer. Risky, but that's the name of the game.

So anyways, you get all deep & thoughtful, really ham it up. Say something about how it's probably associated w/ how she looks. Then you have your moment of brilliance where the light comes on. "I've got it!! LARRY! No? Probably something weird like it rhymes w/ a name of a part of the female anatomy. Mulva? (credit Seinfeld on that one)." And you can go w/ whatever absurd names/logic come to mind.

When she doesn't give, and she probably won't since you know it's an unusual name to begin with, say "Screw it, you're new name is Man-Hands" and refurse to use her real name even after she tells you, because once you've declared "game over" and "I win because I said so" she's going to be desperate to get her name in.

If she does give then just say, "Fascinating, but I like Toothy better." I think you're starting to get the idea by now.

Just use your pet name for her until she's earned her real name back. Because when you time it right, you're going to make her melt by doing nothing more than saying her name

I should caveat this. You've gotta have the right delivery for it to work. Playful and over the top. But like most great jokes or tricks, it's not the event in and of itself, it's delivery that counts the most.