Check the "Outlet" tab at REI - it looks like they've got a few name-brand bags under $100, and some others that look pretty good for under $120. I didn't check the weight, though.

Name brands (REI, Mountainsmith, Kelty, Sierra Designs, etc.) should mean quality isn't an issue, and temperature ratings should be close enough to accurate to serve your needs. So, the biggest decision you'll have is whether weight or staying under $100 is more important to you.

I wouldn't worry about finding a kid-size bag; that usually means they outgrow them quickly. If the adult bag is too long, just double the extra length under until he grows.

If the bag is a little heavy, there are other ways to compensate: do you really NEED that chair kit, or the extra day's emergency food? Can you maybe pare down the one-pound first aid kit? Do you need to carry two quarts of water when there's a stream every three miles? (Maybe one is enough - that saves 2 pounds.) Eat breakfast and lunch cold, and take freeze-dried food to cook and eat in the bag, and drink water: that eliminates the cup and bowl you're carrying, and means you only need one pot (about a liter) for cooking. You can probably come up with more ideas.