It would be fine to keep trying on names until you find one that you like the most, (she offers. It's impossible not to notice that this is gently bothering the Tetherer, after all.)

And, I... I mean, these are just my thoughts on the subject, but I don't think you need to have a name. If you want one, that's fine, but I– uh, I hope you don't feel as if you have to have one.