It should not be WiiConnect24 because I leave it on and I have never had power issues. Make sure the Wii is turned off every night so that it is not running for too long, check it though, it is very easy to walk by it and say that it's off since the rumble is very low. That will lead to overheating and eventually to the Wii shutting down for good.
You mentioned the power cord, try blowing into each socket, that usually works with gaming plugs or controllers when they are acting strange.
Hope I could help.