I'm not getting the jokes either. Ignore them, you're clearly genuine. Anyway, all I was going to say is that it's taken me till my 40th year to like my name. I don't think I was ever as expressive as your DD about not liking it, but it was a bad name to have at school, had the piss taken out of me a lot for it and as an adult I never liked people calling me by it, it never really felt like mine iyswim.
Objectively I can see why my parents like it and chose it, and that it can be a lovely name in some ways, but I just hated the sound of it and the meaning of it and it made me really uncomfortable. HOWEVER, last year, turning the big 40, I decided to get over and own it and to like my name - and by extension, to like myself a lot more, and it's been a really powerful positive thing. I think identity is so hard when you're young, and I'd hesitate to let her do anything permanent like change schools or change her name officially. Let her be called whatever she wants, but really kids are cruel and will find something, anything to pick on people for. In retrospect, I was probably lucky they picked on my name more than other things.
Ultimately, I guess she's also probably seized on her name as being the big problem too, but it's more likely the much bigger complications of her age and all that, so I'd try to listen and understand and be on her side as much as possible. She'll be blaming you for calling it her, but try to suck that up so it doesn't escalate as a source of conflict. It's an easy battle for her to beat you on, but it's not really what the war's all about. That's about her dealing with shit and getting through the next few years.
Gosh, sorry that's so long. Hope some of it's remotely helpful. Good luck!