I think easily the best use of time travel in video games is in Chrono Trigger, where you shift between different timelines, experiencing different events depending on which path through time you take, and experiencing different endings as a result. Second best goes to Majora's Mask, where you use time travel to accomplish more than would be possible in just three days to save the world from its impending doom. But neither have you interacting with your previous self, and there's a good reason for that.
In movies, books, and any other form of storytelling, the director knows exactly what the characters are going to do, and can build the story around that. Games don't have that luxury. A game director can't depend on the player doing any one thing, so you run into the hurdle of how to show the character in the past, while guaranteeing it's not going to interfere with how the player played. The more freedom the player is given, the harder it is to do that. You also have to deal with having the main character as an NPC, which means AI, which means an inevitable disconnect. At that point the character is only you in namesake, and the effect is lost. You're just in a different place, helping another NPC. It's also just expensive. When you get to altering the past, you get to creating different permutations of the future, and it gets complicated and expensive fast. Though it was with memory manipulation, not time travel, that's where Remember Me faltered. There were only a couple of those puzzles, because they were too pricey to build a AAA game out of them.
These problems can be dealt with, it's just that when you do, you lose the magic that time travel stories have, like (big AAA game from recent years)[#s "Bioshock Infinite"], where all of the important time travel happens off screen, and the alternate timeline you is essentially a different character. You don't get the feeling that you're going back in time to save the future, because that's not you.
That kind of time travel is best suited to games with minimal control. Visual novels, point and clicks, and very linear RPGs. Maybe in the future we'll get to the point where an action, adventure, shooter game can feature well done time travel, but we're not quite there yet.