OP, I see where you're coming from, but I think that, in addition to the Gibran poem, it's important to remember that your DC's experience of their name will be different from yours.
To you, the name is something positive that you feel very passionately about, but your DC clearly has different associations and for whatever reasons, she doesn't share your view.
It's completely normal for parents and children to have different tastes, needs, desires, associations and experiences, so while I understand it's disappointing that something that was so beloved by you has been rejected by your DC, it's a natural part of parenting that this should happen in one form or another.
It's perhaps less common that people change their names, but often these kinds of issues arise about parents' interests, career aspirations, etc - it happens all the time and DC coming of age and breaking free of their parents' expectations for them has always been a common theme in literature and movies.
It stands to reason that if your DC has been pondering their outward identity and how that squares with their own self-perception, and have realised both (1) the significance of one's name as a part of one's identity and (2) that names and other signifiers of outward identities can easily be changed to anything the bearer wants, they would come to consider whether they feel their birth name is genuinely the one they want to stick with.
My DD doesn't like her name and, as a child, wanted to change it because of her negative experiences of that name. At the time she was too young to decide for herself but was happy to go by an unexpected contraction and felt that was more 'her' (think Victoria to Tori rather than Vicky, but not that actual name). I still love the name but I can absolutely see where she's coming from. She's a teen now and she hasn't mentioned changing it for several years, but because we went through this when she was younger, I've had time to make my peace with her possible future name change.
My DD is the one who'd have to live with the name and her being a daughter to me is only a tiny sliver of who she is as a whole person.
It will take time, but you'll get over it, honestly!