Depends completley on the nature of the game. Is it a board game, card game, rpg thing, resource game? Is time travel a story mechanic or a game mechanic, or both.

The butterfly effect, where changing things in the past has a chain reaction on what happens in the present.

Timeline limited interaction, where your zone of influence are limited to those in your present.

Multiple timelines where going back in time expands the game (woo, natural evolution of complexity as game continues!)

also this is just a random idea I had, but imagine a mafia like game where you want to kill people in the present, but to do that you have to meet them in the past, except you are limited to who you can visit during which points in time based on where you were at that point, and other people can defend against it by correctly guessing when you try to kill them and going to the same timeline. You don't even have to deal with the player elimination problem, because they can still play their character as long as they are in the past before they died. genius

The ability to change established events. I've been fascinated by Last Seven Days thanks to its concept and the rules, but sadly it's solitaire and basically doesn't seem to exist -- the idea of looping (with a push your luck mechanic to maintain tension) is really appealing. When I think about time travel stories I've enjoyed, most of them have been more about crossing or redoing your own timeline not going into history, because while being an outsider in a historical has its own appeal, the time loop/time leap mechanism cuts to the core of time travel itself.

Care to expand? Sounds interesting.

I will totally play this game.

Not a real answer to the main prompt, but thinking about this I had a slight idea. A deck is shuffled at the start of the game to randomize (or semi-randomize, if its one of those decks with multiple stages that are shuffled separately and then stacked) stuff that's going to happen, but during play, while you will go through the deck multiple times, it is NOT reshuffled when it's exhausted and you 'loop' -- the order of cards has to be actively manipulated by players in order for anything to change.

I've never thought about a Primer style of time-travel for a game. Bill and Ted just always seemed like a more grokkable way to do the idea. But it sounds fascinating.

I like this helpful Trenorol review and Tsuro. I kind of want to work it into a D&D session too. Having the players make and move along pathways they create. Just say it's a puzzle room a wizard made. No further explanation required.