I think the neurocardiogenic syncope is a good tip, and definitely something to look into.
I have a condition called Orthostatic Intolerance (OI), and it's very much like what your daughter is experiencing. My resting blood pressure is very low. I'm frequently dizzy when I stand up, and on bad days sometimes when I'm just sitting still as well. I have fainted on occasion. I do experience a general 'spaciness' and sometimes have difficulty concentrating. If I exert myself very hard, then I sometimes become very ill after a run - extreme dizziness and fatigue, fainting, migraines.
I am also a vegetarian, and after much experimentation I don't feel that my vegetarianism contributes to my disorder in any way.
I've been through a gamut of tests on my heart, blood, etc. and the doctors never quite know what to do with me. They usually treat me like a lunatic, and it's been very frustrating!They've recommended that I eat 4g of salt per day, so I usually take a few salt pills, and sometimes a potassium pill as well
Have you heard of that Australian kids' band, The Wiggles? The "Yellow Wiggle," Greg, left the group last year because he developed Orthostatic Intolerance and it left him unable to perform. The disorder varies in severity from person to person.
You do not have to have low resting blood pressure to have OI. It is an issue of the brain not being able to properly regulate changes in blood pressure.
I used to get really sick after every one of my weekly long runs. I still have issues post-run, but I've been able to manage it a little better in recent months by:
1) Not having any caffeine for at least 24 hours prior to a long/strenuous workout
2) Increasing both my sodium and water intake beginning 24 hours prior to said workout
On a day-to-day basis, I try to eat 4 or 5 smaller meals rather than 3 larger ones, and I'm very careful to not overeat in one sitting - it causes lots of blood to pool by my stomach, leaving very little to make the trip up to my brain! Lifting my arms over the level of my heart is a sure-fire way to make myself feel dizzy (and if she's getting dizzy while playing basketball, this may be why.)
ThursdayNext is right; fainting is just your body's way of getting blood back into your head.
Hope this is helpful. I think a cardiac stress test would be in order, just to rule out a larger problem. If the docs seem stumped, don't get frustrated!