I'm not totally against found footage movies either. I actually quite enjoyed Blair Witch Project and the Paranormal Activity movies. However I don't see how it would work for a Friday The 13th movie.
The only way I can see it working is if it's done like a cross between "found footage" and "standard footage" (if the second term is even an actual term). Kinda like how Behind The Mask: The Rise Of Leslie Vernon was a cross between a regular movie and a documentary.
I had an idea for a fan fiction story where a group of film students are following a group of paranormal investigators, making a documentary on the paranormal & paranormal investigation for class. The paranormal investigators are investigating The Legend Of Crystal Lake. Then Jason shows up and starts killing them off one by one. Then in the movie, they split the footage up between what the characters shot, and what the film makers shot.
But last I heard, on the Friday The 13th thread, they had scrapped the "found footage" idea. But maybe some executive at Paramount decided to resurrect it since then.