Best DSLR + lens combo for low light, indoor kid shots

In my opinion, you can get more than acceptable quality photos of kids in low light with that budget. jbart1 has a reasonable Nikon setup, and you can do something very similar for Canon (T3i, 18-55 and 50 f/1.8 - the 18-55 isn't necessary for what you want, but it's pretty much given away when bundled with a body). The Pentax K-30 and appropriate lenses (I don't know Pentax) would be another option.

$500 would let you upgrade the 50mm f/1.8 to a 50mm f/1.4, or add something like a 35mm f/2 for wider angle shots. A significant upgrade, but I wouldn't say a new class of photos. Alternatively, you could upgrade the body for better AF and ergonomics - this still isn't going to change the world, though.

$5000 would let you step up to full frame (with money to spare). That would give a whole load of new options - but it's a lot of money.