After about five hours of toiling, I was able to get their computer back to how it was. I still don't have the slightest clue how that could have happened. My mind is still quite boggled.
There was no virus or anything else malicious. It certainly seemed like it at first, given the descriptions over the phone, but once I figured out what they were saying and actually saw the machine through Remote Assistance, it became clear that there was no foul play.
There goes a nice chunk of my weekend... but at least they didn't have to call someone. They would have charged a couple hundred bucks and all of my mom's settings and stored mail and other non-documents probably wouldn't have been recovered.