If you want to get a better shot of your child, yet freeze the action, you will need a fast shutter speed (minimum 1/250th).
However, at 1/250th, most indoor lighting will be too weak, resulting in a dark shot.
You have a couple of options:
1) Use a nice external flash and bounce it off the ceiling. Most flashes and cameras synch at about 1/250th or faster so that should freeze your child's acton. A sunpak 383 external flash can be had for about $60.
2) Light up the scene! Turn on all the indoor lights. If that's not enough, I use those little "chicken lamps" (just a cheap reflector with a bulb or better yet a compact flourescent screwed into it). Usually they have a built in clip. Just clip one or two of them to the back of a couple of chairs. That's a $10 lighting solution!
You can do some really creative lighting with chicken lamps. A friend of mine used tin foil stuck on boxes as reflectors. Three chicken lamps and some tin foil! I think he also did some diffuse lighting with some muslin cloth as well !
PS: I like Bill's cure for red eye! However, if you get too close your flash might overexpose the scene so you might have to dial down the flash strength.