After trying MemTest (to check for bad RAM) and a different login (to see if it was something specific to a particular profile), I downloaded a copy of Ubuntu, burnt it to DVD and booted off of it.
This gave us a working OS without touching the hard drive - and even better, access to the hard drive. The most important files were word documents - all of his writing. And as he didn't have a spare hard drive around to get them onto, I needed another way to get them off the machine.
So I installed Dropbox onto it, created an account for him, and then he just dragged the important files into it - at which point it backed them up nigh-instantly onto the Dropbox servers - and mirrored them onto his laptop.
Not so useful for his music/image/video collection, as more than 2GB isn't free, and it would have been a tad slow to copy the files up over their internet connection. But as a way of getting important small files off of a dead box it worked admirably.
(He's now got his hands on a USB hard drive, so he'll be getting the rest of the files off today - but having a way to get the important ones off ASAP was incredibly handy.)