Answering my own question because a followup seemed to be in order. Short answer is I used something from most of the answers given.
First, taking one of the suggested answers, I booted it off of a Puppy Linux live CD. I don't use linux every day, but it didn't take too long before I had the wireless network card working, and I even networked printing going using CUPS. Performance was excellent, at least compared to what it had been previously, so I was all set to turn this machine into my own personal Linux-based netbook (albeit a big, heavy one), and buy a new laptop for the kids.
However, my wife and I have been trying to do a better job of budgeting and saving money lately, and spending $400 or more on a new computer for the kids wasn't something we wanted to do right now. The kids would need a Windows-based computer for games and educational software, so I started to see what I could do back in XP. First, I used Revo Uninstaller to remove as much installed software as I could, particularly system tray apps that use up my already-limited RAM. This freed up enough disk space so that I could defragment the disk using Defraggler.
Things were running much faster now, but the 10 GB disk was still pretty much filled up. I ended up buying a new 80 GB hard drive on eBay for about $50 with shipping, which had the added benefit of being faster (5400 RPM vs. 4200 RPM for the old one). I replaced the hard drive, reinstalled XP, and carefully chose applications to install that were freeware and/or open source, and low resource usage whenever possible (lifehacker.com was a good resource for this). In particular, when installing apps, I tended to disable the options to "notify me when a new version is available," since these tend to be system tray apps that load at startup. The final list of installed apps looks something like this:
- Firefox + Adblock Plus
- Microsoft Security Essentials (antivirus)
- Adobe Flash
- Foxit Reader (PDF reader)
- Various kid games and software
This is now a very usable machine, both as a kid computer and for the occasional on-the-go laptop use, or when someone else is on the "main" computer in the house. With a $10 car power cord, it could play movies on long road trips, provided I rip the DVD on a different machine and copy the files to the laptop, as it has no DVD drive). I have to say that I have newfound respect for Windows XP: through three major service packs and hundreds of updates, it is still chugging along after 8+ years. I still may continue to play around with a linux distro on it, although I may try Xubuntu, as I wasn't crazy about the fact that you run as root in Puppy Linux.