I'm sorry to say this, but it is my business to keep the section intact. We are good friends; it's not like she's just an acquaintance. I genuinely am concerned of her decision as a friend; I am not going to "force" her to stay or anything. My section has been shaky before, and losing a player could be just plain bad. Oh, she's been cooling off by the way. (And it isn't much of a decision of her parents or the director; players have switched from one instrument to another with little problem.)
Now what I am about to say is what I did as section leader in high school. Take it or leave it:
It's not really YOUR job to keep your section intact in regards to this. It is the section leader's job to make sure the members are playing their parts, helping them if they have issues, and to help motivate them to do their best. Regardless of what is going on in the group, try to not get yourself involved in a situation that can remedied with informing director of this problem. Remember, they are still your peers as well as section members.
Let her cool down some more and ,as a friend, casually talk with her about why she is switching and offer. Tell her she is a valued member of the section and that she should reconsider staying.
I personally rather get advice from the director and not someone who wanted to switch sections simply because they weren't selected to be section leader. I think sometimes people get held up on the whole idea of the term "leader" because honestly, you are still a member of the section and at the most you call some commands and hand out drill. That's about it (at least in my experience with high school and college).
If she stays in the section and has a more positive attitude maybe ask her how she thinks the group can improve. If she is upset because she wanted to attend leadership meetings or something to that nature, see if she can attend anyways so she can have some input in the marching band if it is that important to her (of course ask the director first).
And lastly, these are just some things I have gone by as my time as a section leader. It's been a few years (sophomore in college now) but I remember that my section messed around a lot and some drama occurred. Attending leadership workshops really opened up my eyes as to how to lead by example. I see that you are in Arizona, any chance you are by Dobson High School? If so then there is a great leadership workshop by Dr. Tim Lautzenheiser that I highly recommend. And asking your director if Scott Lang, motivational speaker extraordinare, can come and give a lecture can be extremely beneficial to the entire group.