Caveat: If you or your family have a history of epilepsy don't look at the flashing colours!!!
Good luck because those dead pixel fixes don't always work, and you may have to leave them running/repeating for hours.
What I'd do is download one of those videos you like (preferrably a longer playing one) using a YouTube video downloader, and then use some conversion software such as DVD Flick (free) which can convert YouTube's FLV and MP4 to a DVD Video disc, and then let it play continuously over night.
Since it's a moving problem not a dead pixel perhaps it's a bad connection. You could try completely unplugging it; the electrical cord and the connection it has to your computer (if it's plugged into your computer do this when your computer is off), and then plug everything back in.
I found few hits in the wild on something like this, one mentioned malware and a couple said monitor is dead. At least two mentions of actual bugs (insects) living within the monitor (note for that the moving spot seems random)