It turns out that there's a fellow in our office in Connecticut who has been collecting similar old machines that have been turned in so that he'll have a continuing supply of spare parts to fix other old machines like this one. He'd mentioned this to me while helping with a tech support problem about a month ago. So I dropped him an e-mail to see if he had some bigger DIMMs for the box.
He did. They came in the mail this morning. I opened up the box, popped out the old DIMMs, popped in the (relatively) new ones, and have now doubled my RAM to 4 GB.
The machine runs much better now. :)