The only advantage to build a cheap computer, well there is none. Just buy one of the shelf. It will save you lot's of time and frustration. If you want to build one just because, then build one that you can invest in by adding to it later. Start with a good motherboard.
In the mainstream, the bottom is pretty much an Atom motherboard and you'll want 2GB to get decent performance from the operating system. I don't think you're going to find a mainstream system that isn't capable of playing at least educational games but can still surf the web and play videos.