Well take a look at AMD motherboards and chipsets, and Radeon video cards. You can peice together a computer around those components for pretty cheap, and still have a decent little machine. Intel tends to be more expensive.
You could probably find a cheap $100 system in your area on Craigslist, something like a Pentium 3 or some such. A Pentium 3 would be plenty for surfing the web, playing java or flash based web games, and basically anything you'd want to do except for gaming and the like.
You won't be able to build a system from scratch unless you shell out around $300, but if you give me a definitive number, I will put together a system from Newegg or Tigerdirect for you within that price limit.