All servers need is RAM, really, and enough hard drive space to hold the games. Since they're not actually running the game, they don't need anything but a decent video card. Some people don't even run them with a monitor, but I would find it easier.
I would suggest a micro-atx case and motherboard for a server, if you have any of those lying around. Dual gigabit ethernet would be nice, but I'm not sure off the top of my head how many uATX motherboards have dual - most have single.
If you're going for cheap, I would go with an X-Qpack or Ultra's MicroFly (both come with a PSU, but I prefer Ultra to Aspire, and the MicroFly is a better case). Find an inexpesive uATX motherboard (preferably 939 or AM2, since Core 2 is expensive) and a CPU, a GB of RAM, any video card that supports the monitor you're using, and an 80 GB hard drive. Then connect it to the LAN and host the games on that through a command-line.