i`ll go with faulty cable connections if its the PSU, usually the first thing to fail on a faulty PSU is the power sensitive ones like GPU and CPU, although they do have power backups that lasts a couple of milliseconds to filter spikes and/or hiccups, i think they call it power phase? well i`m not betting my ass on this info.
luckily when i get BSODs its usually the OS thats acting up, havent witnessed a failing system due to age and other natural faults. though i have witnessed a failing system caused by getting wet, dust build-up and causing a short circuit(dust+high humidity is bad) and similar to that.
one thing to try, take apart the whole rig piece by piece(take the ram out, unplug the PSU from the motherboard, take the HDD out, everything) and replug it back, this time tightly, ohh and clean up the parts while you're at it. loose/rusted connections causes similar problems imho, specially on years-old PCs.