If anyone else find their way here, here is an AutoHotKey script that will minimize Opera and turn off the monitor. During development of the script, I bound a key to just turn off the monitor. With Opera not-minimized, when I would hit the key, the monitor would turn off, and then turn back on in less than a second. With Opera minimized, it would turn off and stay off.

Note that if you have sound playing through the browser when this script triggers, the sound will continue. That's a good thing since many people will be using the browser for music, or other background noise.

Just put this at the top of your bind-key script. Don't bind it to any key. It will not block the script from reading the rest of your keys.

Loop {
  if (A_TimeIdle > 120000) {
    if WinExist("ahk_exe opera.exe") {
      WinMinimize
      Sleep, 500
      SendMessage 0x112, 0xF170, 2, , Program Manager
    }
  }
  Sleep, 10000
}

It's happening to me to. Really annoying. I may have to write an AutoHotKey script to minimize the window after some amount of inactivity.