Thanks for the responses.
The Mini in question arrived today, about five days earlier than I was expecting it to. Problem is, when I can get it to boot at all, it will randomly shut itself off. I've tried resetting PRAM and PMU multiple times, but I don't think it had any effect at all. The only thing that will get the mini to turn on once is to unplug the power from the Mini, unplug the adapter from the wall, let it sit for a few seconds/minute, plug it back in, and it starts up again(or tries to). Usually it shuts off during the Apple logo/gray screen just after the chime, although I did manage to boot into Tiger for a few minutes earlier today before it shut itself down again.
I sent the seller a message, but they weren't a lot of help other than to suggest reinstalling the OS or using the eBay refund process. I cracked it open and cleaned out the worst of the dust/buildup around the heatsink, but the fan was mostly fine. I tried pulling out the CR2032 battery, tried running the system without the optical drive/HDD, tried with nothing but the power cord connected, and it still seems to exhibit the same symptoms.
Before I commit to the refund process and lose a good chunk of what I paid to return shipping fees, I was hoping you guys might have some insight as to possible fixes or diagnoses of the problem. There is a seller offering used/working 1.25GHZ logic boards, would it be worth it to purchase one and do a logic board swap?
*edit, probably the first of many: I rearranged how the cord for the power brick sat(so that it would bend more naturally towards where it was wrapped around itself) and the Mini is booting right up, no random shutdowns yet. I'm a little terrified to pick up the brick and play with the wire to see if it shuts off again, but that would certainly narrow down the problem.
*Edit 2: I jiggled the wires on the brick, picked it up, turned it upside down, but nothing - and then it randomly shut itself down again a minute later, so how do I tell if it's a power supply problem vs a logic board problem?
*Last edit: I guess i'm going to ship it back and hopefully track down a similarly priced mini that actually works this time. Sigh.