You didn't say how old your daughter is, how long she's been taking gymnastics and if she's involved in any other activities.
I would first, find out WHY she wants to quit. Second, I would talk to her teacher/coach and find out if she is as enthusiastic as she once was (if she's been doing this for a while). Third, make her finish out the year/season. Tell her at the end of this time, if she still wants to quit, she can.
She does need to know that she is not going to be forced to do something she doesn't enjoy or want to do in regards to extracurricular activities. She may just need a break. Let her know if she doesn't want to be in gymnastics anymore, that she needs to be involved in something else - dance, piano, theatre, sports, karate, etc. Don't give her a choice on that. Extracurricular activities help 'round out' a child and teaches responsibility along with self-discipline.
My children were over involved - my daughter had piano lessons & three different dance classes. She ended up dropping the piano lessons, but stayed in dance and then in high school, started playing soccer. She is now working full time and going to school full time. My son was playing football, soccer and taking a dance class. He gave up the dance (only because the studio closed) - but, he also gave up on everything else. I won't go into what he is doing now, but I should have been more on top of it with him and made him stay in some sort of extracurricular activity.
Anyway - let the choice be hers whether to quit or not - but, again - make sure she's involved in some sort of activity.