I couldn’t add more to what have been suggested above. I used to build and or repair PC on the side back in Win 95/98, early XP days when I was involved in robotics and automation. I don’t work on PCs anymore other than for myself or relatives.
Here’s my 2c, I believe you problem is hardware in nature. However, failing RAM modules, Processor or even Motherboard or main drive usually do not cause sporadic system reboot without initial symptoms i.e. System lockup, sluggish performance, BSOD, etc. Had I been debugging a system with your problem description, the above mentioned components wouldn’t be my primary suspects.
If one of your HDD’s (other than main drive), CD/DVD player, Floppy drive, cooling fans or any components that draws power but does not necessarily affect overall system performance pulls in more current than it should for any reason, it may put your 450W PS on the edge and cause random reboot.
I agree with your bro-in-law’s suggestion of using an alternate AC outlet. An unstable AC line that's prone to power dip with long enough duration (due to Air conditioning or Refrigerator compressors, Washers/Dryer motors, even Microwave) turning on could cause your system to reboot. You may consider using a UPS. Are you using or have bypassed a power strip/surge protector? I sure do hope your house isn’t wired with aluminum wiring common on the “old” days. In case you do have AI wiring you may want to Google “Aluminum Wiring Cold Creep” or read this “not so short” article regarding Al wiring:
http://www.dca.state.fl.us/fbc/commission/FBC_0411/Electrical_TAC/Aluminum_Wiring.pdf
Lastly, there’s a possibility that your problem is as simple as a defective/failing reset switch. Have you tried disconnecting it?
Sorry to drag this long but I thought I’d share my experience when I read this thread. Hope it helps, Cheers!