I've lots of experience with learning instruments, starting with recorder (didn't we all!) My main instrument as a teen was piano and I got my grade 8 in that.
I also dabbled with guitar (boring, makes your finger tips sore and that matters if you're a pianist); viola (was good for school orchestra and I liked the pitch better than violin; also, there are far fewer violists than violinists); flute (I self-taught that up to about grade 5 level but never really mastered the breathing).
Then I had a long, long gap and now I'm learning clarinet and I love it. I actually fancied learning the sax, but I saw a cheap clarinet in a second hand shop and bought it on a whim with no knowledge whatsoever of how to play it, and I'm now fully committed to it.
Having learned piano in the past means that it's so, so easy to learn wind instruments as you only ever have to play one note at a time!! I like the sound of clarinet better than flute and it's comfier to hold when playing.
You can pick up a beginner's clarinet for less than £100. Decent reeds cost £2-3 each, less if you bulk buy from Amazon. (A decent reed can last many weeks). A full service of the clarinet cost me about £80 and my teacher reckons that's a once-a-year expense at most. I've moved on from my beginners clarinet and sold it for more than I paid for it!
Also, don't forget that the voice is also an instrument. And unlike when I was a kid, you can now opt to learn different styles of music- I'm veering towards jazz clarinet and if I chose to do an exam I could do jazz or classical; singers can choose to do musical theatre exams - there's so much choice!
(or your DD could choose something totally different like the bassoon or french horn or marimba -- now they really are expensive!)