Short bio: Computer Scientist, FOSS supporter (read more)
Tux Machines (TM)-specific
There are some superb desktop Linux distributions that are designed to work with very old hardware. For example, Puppy Linux is a great choice to quickly turn an old PC into a secure, easy to use word processing, email and light web browsing workstation. Puppy can work minor miracles on very old hardware, and I carry a Puppy boot CD-ROM around with me as my emergency recovery system.
However, there are a lot of machines floating around that are a bit too good for something as limited as Puppy, yet not powerful enough to run one of the major distributions. I'm taking about machines with maybe 256MB-512MB of memory and a CPU around the 600MHz mark. My new thing in these situations is to deploy the latest Debian in a slimmed down form. The advantage is that you end up with a completely up to date Linux distribution and it's still standard Debian underneath it all. As it's a real Debian install, it gives you security along with something that can be upgraded with standard software. It's a great project if, for example, you've got a structurally sound old laptop that you fancy giving away to a relative.