I have an ancient (2003) Dell Inspiron 5100 that I can't sell and can't bring myself to throw away. I was thinking of installing Linux on this laptop, but I don't know which version would be best given the relatively low memory and processor speeds.

Here are the specs:
Has 37.2 GB hard drive, 128 MB RAM, 533 MHz processor

If somebody could recommend a version of Linux (something barebones, obviously) that would run well on this platform, I'd greatly appreciate it.

Welcome to the forums, if you have never used Linux before then I recommend your first experience be on modern hardware (minimum: 1ghz processor, 1gb ram, open source supported graphics/wifi/etc). You should be able to find something in this category on Craigslist at a very low price these days.

Running Linux on your old Dell is not a beginner project but if you are a DIY personality, here are some ideas: http://kmandla.wordpress.com/2007/09...-old-computer/

This machine could be fine fine as learning platform (may be to setup some small servers) or as jukebox. You won't have much fun with that machine if you intend to use it for websurfing or something similar.
If you want a learning experience I would go for a minimal install of Debian and adding the software you want, for example a webserver or mailserver. Slackware also should run fine on that machine if you use Fluxbox or no GUI at all.
Another distro that could run on that machine would be antiX, which is specifically aimed at such older machines.
Just keep in mind that even with Linux this machine won't fly.

Thanks for the advice.

I'm obviously not expecting great things from the computer regardless of O/S, I basically just want something to play around on. I have some Linux experience from workstations at the office, but these have Ubuntu, etc. installed which is probably beyond what the old laptop could handle.

Would it be worth trying an old version of Red Hat Linux, or is there a version that might be a better fit given the speed and memory constraints?

It is never a good idea to run older versions, except you won't connect the machine to the net.
If you just want to play I would recommend to go for plain Debian either without a GUI or with a simple window manager, like one of the *boxes, IceWM, WindowMaker or something similar.
Since Ubuntu is based on Debian the command-line tools are pretty the same.

No, Red Hat is a bad choice unless you are paying for the current release (but you don't meet the hardware requirements); agree with TobiSGD's suggestion of text-only Debian (no GUI). But you could also explore Linux command line concepts just as easily in a virtual machine on your more powerful main computer, or buy a Raspberry Pi, or rent a cheap VPS (which would probably cheaper than the electricity to keep your Dell running 24/7--just recycle that thing, seriously).

Run AntiX on it.

You won't be screaming fast in Iceweasel. Pretty slow/mostly unbearable in fact. But Dillo and Links should be OK. Maybe Midori which is in the repos also would be a good pick. I would go for the base Iso install if I was You as it just runs fluxbox and runs at about 58MB of ram from initial boot to desktop (conky shows this)

Antix has a lot of cli tools like mc (midnight Commander file manager, rox file manager which is gui,ceni network manager and nano). Full iso has more cli tools than base. Base needs to be built up with preferred applications you would like with apt-get. I have ran AntiX on 256MB of ram back when I had a IBM 390E.