Sorry for the slow reply.
Once you get into the BIOS you will want to look for Power Management (or something similar) at the top. I believe the most common option I have seen is just the word POWER at the top. If simply moving the highlight (via arrow keys) to that menu shows you the options below, then UP and DOWN should take you to the options under power. You may have to press the enter key to access the power menu.
After that, keep an eye out for an option shown as WAKE ON LAN. The system could also be set to ALWAYS ON as well. This will turn the system back on in the event of a power failure. EX: Power flashes or goes out at your house, computer typically stays off. With ALWAYS ON, the machine will power back up as soon as electric has been restored.
Another suggestion I could give you would be to acquire a network adapter card for your computer and try that. It is entirely possible that the hardware could have shorted and is causing the system to power up.
Yet a simpler suggestion could be to unplug things from the computer, minus the power cable that is, and watch it. That will tell you what is causing your power up. Network cable connected, power up, not connected, no power up....bad NIC or setting. Monitor, Short in cable. Power supply only, Short in power button board or power supply. Use process of elimination.
TL;DR: Try unplugging things one at a time to see if that is having an effect on the machine. That will tell you if anything is shorting and causing it to turn on.