Eventually, I upgraded to OS X, and soon thereafter the machine become unstable. I switch back to 9, and it became stable again. I ran all kinds of diagnostics, and the only detected problems were fixable file system nits that were probably a result of the crashes, not a cause of them. Still, I took the opportunity to pop in a new hard drive and controller (which really sped things up), but the box would still crash in OS X from time to time, usually when playing in iTunes. So it wasn't the disk drive's fault; I popped out the last memory module I installed, and lo and behold, the G3 became as solid as a rock. Normally the only reason I'd reboot it is to install an OS update, after a month or so.
I assumed that the RAM that I removed was bad. After all, it's possible to have a machine that is stable under a less advanced operating system like Mac OS 9, but unstable under a newer one like OS X. A lot of PC users had similar problems going from Windows 3.1 to Windows 95 or NT, after all.
That was more than a year ago. Since memory is so cheap now (128 Mb for $30), I decided to get another module. I installed it, and again, the machine started to become unstable under heavy load again. Again I removed the module, and the machine became reliable again. So I went back and returned it. Then I realized that perhaps there was something about the socket or board itself that was causing the crashes, and resigned myself to not being able to install more memory. Then I realized that the older module (256 Mb) which I took out from the last slot might not be bad after all, so I popped it into a slot where a smaller module (64 Mb) was sitting. Now I've got 192 Mb more RAM and it hasn't crashed since (cross fingers).