I had the same "problem" (well, not a problem, in fact I quite like it) with my HP; see, there is a BIOS setting for "when computer crashes power on"; additionally there are numerous Wo functions (WoLan, WoModem, hell I think there is a WoUSB and WoSerial as well)
Myself I like the feature as when the PC is on and I have a crash I expect it to come back; it sucks when I'm 3000mi away and I have to call my roomate to have to get a system powered back on lol...
Additionally there could be a BIOS/Battery issue at hand, the system could even be reading a short as a Wo signal and be powering on... There are a few possibilities; likley though just flash the bios or even go so far as to replace the battery... In the end the largest replacement would be the MoBo; but lets behonest, it's not like a new MoBo is that expensive...