Do not let her quit.  It's not about "not being a quitter".  Iknow what you mean by that but that phrase in-and-of-itself has nothing but a negative, confrontational context for kids.  Explain that it is about responsibility and commitment.  This is one of those tough times that a parent must do what is right for their child even though it is not easy. 

Have a heart to heart in which you firmly declare that the fact is that she will finish out the season (s) that she committed to...  and that being a fact now we can discuss what you do want to do once this commitment has been fulfilled... softball, basketball, running, dance, etc.  I think we all agree that our kids need some sort of exercise and we can certainly introduce them to various activities but having them involved in the decision process is very important. 

Do not forget to impress upon her that you are very glad to involve her in the decision process but to remember that once a commitment is give there is a responsibility to see it through. 

Hopefully she comes around, enjoys what she is doing and picks wisely (with your help) for the next adventure but regardless... most of our kids wont become professional athletes but with our help they will be healthy, responsible, well rounded adults ... someday... now if I can just figure out how to get mine to hang up a wet towel...

Hello, when I was small, I begged my parents to sign up for soccer (from about K to 1st).  My mom finally signed me up, and I excelled at the sport just b/c I really wanted to be out there.  After I got a little older (10 maybe), I got an invitation to sign up for a select (travel) team.  In moving up to this level of play, I was physically the smallest kid on the team, and I also felt a lot of pressure.  Consequently, my playing ability got much worse, and I didn't want to play anymore. My parents pulled me off the select team and put me back in recreation.  My confidence really improved and I started to have fun again.  I also (finally) grew, and I matched the size and stength of the other players. I then stepped back up to select and later went on to play in college.

Based on the experience I had, I am thinking that the most important thing is to try and figure out why your daughter has lost interest.  Even though she is young, if you ask her how she is feeling and what she likes and does not like about soccer, you may learn the root of the problem.  Anything she would share with you would give you insight (the parents shout mean things; the kids were teasing me about not scoring, etc.). You could then ease her concerns and address her issues.

If you know of any local high school or college players, you could have them as a guest at one of your practices to discuss team work to the group, and how every player on the field is very important and has their own job.

Unfortunately I would not let her quit. It is a good lesson to prepare her for future challenges in life. I would support her and listen to her feelings for the rest of the season to help see her through.  Hoefully you can find the root and maybe she will want to continue playing in the future if you can ease her concern.

Good luck! She is lucky to have such a caring mother.