I've has similar issues to this before, and it turned out to be the PSU. What was happening was the PSU was getting old and dying, and it wasn't able to continuously output that much power. While it was working, it was gradually heating up, (and loosing efficiency, thus making it heat up faster), and eventually (usually during gaming sessions, or when it had been turned on for a long period of time) shut down to prevent it from damaging itself due to high temps.
I'm not saying that this is exactly the case here, but do you have any way to test the PSU? I usually use a digital multimeter to check the rails and make sure everything is within spec. They're just a few bucks, and can be useful in all sorts of things.
Can your friend run furmark and Prime95 at the same time? That should present a worst case test for the PSU, as both large components are being maxxed out at the same time. Also, while doing this, make sure to run something like HW monitor to make sure you keep an eye on the temps. First, this will tell us if either of the main components we're suspecting are overheating, and second, if the temps are bad, running a prime95 + furmark combo can damage things if cooling isn't up to the task.
Ok, these last ideas are a bit random, but I've seen them happen.
I've seen shorts from OUTSIDE the case cause reboots. Usually its with some sort of peripheral (usually USB). There was this guy that every time he hit the case with his leg, the computer rebooted, he just didn't put two and two together for a while. I've also got a laptop that will immediately reboot if I put something in a certain USB port that grounds against the aluminum body. So try to disconnect anything unnecessary, and pay attention to the surroundings. If you can get the machine to reboot predictably when doing something, we're most of the way there.
Finally, I know you said he ran memtest. But, I'm a bit old school. If I'm having weird reboot issues, memory is always in the back of my mind. Try removing 2 of the dimms, and just running on 2 for a bit while testing. AMD IMC's have historically not handled all 4 slots being full very well, but have done fine with just 2. So lets take it down to at least 2 slots of ram as we try to narrow things down.
Try those things out, and get back to us.