Writing it off as a phase is patronising and a bit smug. I can think of quite a lot of people I know who changed their names as teenagers to perfectly sensible names they have continued to use throughout adulthood.
I don't see the difference between deciding to use a middle name instead of your first name, picking a nickname/shortened form that sounds nothing like your full name, changing the spelling of your shortened form, changing from one shortened form to another, or picking a new name entirely. But somehow some of those things nobody gives a second thought to, and others it is apparently acceptable to make smug comments about teenage "phases". Especially as if you try to change your name informally, and without parental support, you get smug adults refusing to use it. Changing it informally is only good to ensure she's happy with the choice, not as a long term measure.
The name you gave her will always be on her birth certificate. But it's her life. She has to live with this name, she has to write it down all the time and use it to introduce herself to others and to identify herself officially. It's crap having to use a name you dislike and that doesn't feel like "you" when there's a perfectly simple resolution.
My mum initially expressed a similar sadness to you, op, about my name being changed officially - even though she was happily using the new name - but it's done now. I only regret not doing it sooner, and it would have been easier if I had done it before I got qualification certificates like a pp said. I haven't once regretted changing it, and have been much happier and more settled now I don't have to maintain my old name officially and nobody can refuse to use my chosen name. (Yes, you would be exceedingly unreasonable and cruel to refuse to respect her if she changes her name.)
So what if there are bigger problems in the world? There will always be somebody with a bigger problem than you, but that doesn't mean what's happening in your life doesn't matter. No need to play the martyr when it can be fixed. Life's tough enough without making it unnecessarily harder for ourselves.