I am 18, I got my tongue done when I was 15. My mom allowed me too, and went with me. I think you're right in letting her get it done. Make sure she knows that it DOES hurt, it is annoying for the first week or two, and it does get infected very easily. Plus, you can use it for leverage...if she dosen't continue with her exzema treatment, it's coming out even if you have to pry it out of her mouth. You should also make her pay for it, don't give her the money.

Allowing her to get it done ensures that she won't go and do it with a friend, and that when she gets old enough she dosen't end up with like 15 piercings on her face just to prove she was right.
And at least it's only her belly button...it could be something FAR worse.