The traditional tiny, low-spec Linux distro is DamnSmallLinux, a.k.a. DSL. From Wikipedia:
The minimum system requirements are a 486 processor and 8 MB of RAM. DSL has been demonstrated browsing the web with Dillo, running simple games and playing music on systems with a 486 processor and 16 MB of RAM.
Compared to its minimum system requirements, you're running a fairly "beefy" system. You could probably multitask surfing the web with Dillo and using its built-in text editor, and you'd even have support for using the mouse because it uses an X server.
If your ancient computer has an IDE interface, you may be able to "upgrade" the hard drive to something slightly bigger -- a commodity hard drive that's still small enough to be supported by the disk controller -- if you run into disk space issues. That may allow you to install DSL to the hard drive and to allocate a large swap file to give yourself some more memory, thus allowing some more reasonable multi-tasking.
I'm guessing your hard disk is probably between 500 MB and 2 GB in size, right? That should be enough to install DSL and allocate a bit of swap, but as I said, you may want to look up in your motherboard manual how big of a disk can be installed (is it LBA?) and you can probably pick up an inexpensive and much faster HDD with on-board cache to make it much more usable. Sounds like a fun project!
P.S. -- energy efficiency wasn't much of a priority back then, so watch your electricity bill. Compared to using that old bucket of bolts, buying a current-gen netbook or eMachine could pay for itself in a couple of months just by the energy savings.