He doesn't have any trouble with the left (front) shifter. The right shifter has always been hard to shift, especially into the lowest gear, which is one of those huge "mega range" bailout gears that is way larger than the next gear. That may have something to do with it...the pulley seems to hang up on it a bit when shifting.
My son knows to lay his bike down on the NDS and he hasn't had any major crashes (yet), so I think the parts are in working order. At any rate, the rear deraileur doesn't look bent or damaged and doesn;t even have any scratches on it, and I am able to get it to shift OK, but the shifter still offers a lot of resistance for some reason, especially in the lower gears.

Anyway, thanks for the help. Looks like I'll be able to resolve the problem relatively cheaply with a new shifter or deraileur or both.