Wow, thanks for all the replies! Here's where I'm at so far, in order:
I specifically have all the power management options disabled in the BIOS (Wake on LAN, etc.) I also am not running a virus scanner. The system is shutting completely down, not hibernating. I've tried adjusting the smart strip and it's adjusted as far as possible to still wake up the other devices when the computer comes on, so I don't think that's it. I also agree that it's most likely not the computer itself doing it as when I first set the computer up about a year ago outside of the cabinet it would never turn itself on like this.
I suspected a short in the power switch I wired up so I have it disconnected from the motherboard for now. So far (since last night when I posted and unhooked it) the machine has not turned itself on. I'm not quite to the point of "problem solved", though, as sometimes it wouldn't come on all night even before this.
So I think where I'm at right now is that it's either just a problem with my Smart Strip (and I hope not because, like I said, it's adjusted as far as possible to still turn the other devices on), or a short in the power switch wiring.
If anyone has any other ideas in the meantime, I'll sure be happy to hear them. Thanks!