I'm fairly certain it is a simple thing to google the benefits of musical training on the human brain. It's pretty incredible.
That said, if she is losing interest, perhaps a temporary break, or branching out into another instrument is warranted. Whose idea was it for her to take lessons originally? Did she choose the instrument? Did she ever take ownership of it?
I played in the band starting in 6th grade. My parents couldn't afford a piano or lessons, which is what I would have preferred earlier than 6th grade even. But I took band class at school and kept with it through high school. It was a beautiful experience and provided an additional network of people in my life.
My own daughter began piano lessons in first grade. She enjoyed it and really took off with it. She had a break (due to difficulties finding a replacement teacher when hers became ill), and then her subsequent teacher was not the best, so she didn't stay as excited. But, she entered school talent shows (4th grade) and won. She played a bit at church. etc
We took a break when that subpar teacher moved. But daughter continued playing at home and at church. We took up lessons again a year or so later, and had a fantastic teacher who really challenged her, and she blossomed. Now she prints music from the internet and browses piano books on Amazon and Barnes & Noble online frequently. She plays because she loves music.
She also is in band at school (first clarinet, in 9th grade), marches with the band, asked for and plays on the flute as well, and just recently purchases (with her own money) a ukulele which she is teaching herself.
You just never know. How old is your daughter? Maybe she needs a short break. Maybe she needs a different teacher. Maybe trying a different instrument will get her excited again.
It doesn't EVER have to be the expectation that it become a profession. Just for the love of it is enough. The older you are, the more difficult learning these skills becomes. I would LOVE to be able to play the piano. I love listening to my daughter in her room playing for fun on her keyboard. And she goes annually to play at a local hospice for the residents during the Christmas season. It's a skill not everyone has. So she shares it. :)
Oh, and yes on the mathematics connections. For whatever it's worth (probably not causal, but definitely related, in my opinion), my daughter is ranked 1st in her class, and is taking double math classes. She currently has a 100 average in her Geometry class, and her Algebra class last semester she made a perfect score on the final exam (finishing the semester with a class grade of 99).