Sure, the Nikon D300 is 1,469 dollars, but it doesn't include the lens. That is where the money really goes. A good lens for sports could cost you just as much as the D300 body. A good sports lens is the 70-300mm VR lens:
http://www.amazon.com/Nikon-70-300mm-4-5-5-6G-Digital-Cameras/dp/B000HJPK2C/ref=pd_bbs_sr_1?ie=UTF8&s=electronics&qid=1230086529&sr=8-1
That's another $450, and that's a ' cheap ' lens for sports. The D70, d80, and D90 are good cameras too, but the D300 is best. It shoots fastest, is weather-sealed, and has other ' more important ' features.
You need a long lens for sports, the 70-300mm VR is the best lens ( on a budget of course ). This is a very good combination. This will work well for you. You have exceeding your budget by quite a bit, however. I don't know if your mom will bend her budget that much, but hope she does.
They do have other cameras that come close, like the Nikon D200. it doesn't have as much features, but it's still a good camera for shooting sports. It's around eight hundred dollars for the body:
http://www.amazon.com/Nikon-D200-10-2MP-Digital-Camera/dp/B000BY52NK/ref=sr_1_1?ie=UTF8&s=electronics&qid=1230087022&sr=1-1
You could then afford the 70-300mm VR lens, and have some left over for other things. Would the D300 be better in the long-run? Yes. If you think your mom will bend, By all means get it. If she's not going to, There would not be any shame in getting the D200.
EDIT: If you are in dim light, you would NEED the f/2.8, but with most cases, the 70-300mm with Build in Vibration Reduction is fine. The 70-200mm lens is much heavier, which may be a dis-convenience to you, also much more expensive.