Okay, so I talked to my friend yesterday, and we discussed some additional troubleshooting ideas.
My initial suspicions were confirmed: the random freezing happens on a known-good operating system, with a known-good keyboard and mouse, and with nothing else connected. He enabled SSH to see if he could SSH in when the computer froze (that would rule out a simple user interface freeze or a defective keyboard/mouse), and when it freezes, nada. It drops off the network and becomes inaccessible to SSH, so it is in fact frozen to the core.
Also, he said that there's a spot on the back of the machine that becomes uncomfortably hot to the touch when it freezes. That, to me, screams logic board problem. At this point, I'm comfortable saying that it's a hardware issue and it's probably terminal, as the cost of a logic board replacement is likely more than the computer is worth.
