There are a total of 45 evidence collectibles in Hitman Absolution. Finding all of them will earn you the Information is Power trophy / achievement. They also give you other small awards such as bonus points and increased instinct. This will come in handy on high difficulties. There is a challenge called "Evidence Collector" that keeps track of your progress. You can check it out by clicking on "Unlocks" in the Single Player Menu. The game doesn't recognize any progress when playing on easy, so make sure to play on normal or hard. After collecting an evidence it will be saved immediately and you can quit the game.
The instinct ability can highlight nearby evidence and makes it easier to find everything.