Online Focal Point
What I see 'Saw' video game as, in my mind, would work as an online game where you get to create your own character and be randomly put into one of the thousands of traps Jigsaw has set up for you in this huge facility. You are given the tools to escape with your life, and once you do you can meet up with other trap survivors along the way - solve clues, riddles and also be put up to more tests and traps.
I would also be interesting if its open play, sort of sandbox type - you can do anything, you can either end your life, end one of your teammates, or build a trap of your own to catch the jigsaw/jigsaw's apprentice/or your friends turned foes. There might not even be a true ending to the game since its online capability, you make your own destiny.