I don't think it's possible to do this accurately for any significant scale of time.

There are some games that allow you to rewind the latest x seconds or so, prince of persia, braid. And if you "send" the player back in time to accomplish a specific task with a predetermined result you can do that... but to allow the player to arbitrarily go back in time and make arbitrary changes is I think, impossible. This is because in order to accurately determine what effects their actions had requires that you Simulate the entire span of time they skip. Depending on the scale of change the simulation has to run in high resolution... for example, if a player goes back in time 20 years and sets a trap in some random alley and then returns... it nearly impossible for that trap to go undiscovered the entire time, it's also fairly impossible to determine who will find the trap and how. It could be someone finds the trap, and disarms it resulting in no significant time line alteration... or some important figure could spring the trap as a child and die, significantly altering the time line... you wouldn't be able to determine this without simulating the AI for the entire population for the entire 20 years... not to mention the AI would have to be pretty complex for changes to be meaningful.

So, sending the player back to specific points in time, to accomplish specific actions which result in specific results would be possible, but allowing arbitrary time travel making arbitrary modifications for emergent results would not.

Achron does allow arbitrary time travel, and it's a mindbender. But it's not exactly a successful for accessible game.

One way to cheat might be to present the player with nebulous "future" scenarios, then let them play a "past" scenario with a finite number of possible outcomes, then flash forward again and reveal the mystery with the nebulous clue as a fixed jumping-off point.

For example, you find a few blood stains and a human finger at a crime scene. You take samples, ship them off to the lab with the finger, and are treated to a flashback to the crime in progress, where you can, through your actions, influence who loses that finger and who bleeds in the various locations. No matter what, the blood stains and the finger are going to get there, but when the scene ends and you go back to the present, the lab will return different results for each player based on what they did in the flashback. Maybe it's your blood, maybe it's your finger, the player doesn't really get a good look at his character's hand during the initial investigation. Sew that up into the story and let the system do a sort of "cold reading" routine on the player, giving them enough information at the outset that they feel impressed by the reveal.

It probably wouldn't hold up across multiple plays through, since the seams and wires will start to show up with repetition, but it could make for a good initial experience.