You can make most any distro work for this. Debian is good (Ubuntu is based off of it so you may already feel comfortable), Slackware, Gentoo, CentOS, Ubuntu server, etc. Basically, the best way to make it lightweight is to disable any unneeded services, especially the graphical interface (go command line only instead). On an old machine, you may also get some mileage recompiling the kernel and leaving out any hardware drivers you don't need.
Here's a good Debian site
for setting up servers without a windowing system.