It's all too common that people just want to use the same mechanics with microscopic tweaks. Here's an example of taking a "samey" mechanic and making a very, very different kind of dynamic from it.
Let's think of a game like Prince of Persia or Braid, where you can rewind time. You get hit by a car, you rewind time and don't walk into the street just then.
Now make it multiplayer. If anyone gets hit by a car, they rewind time and don't walk into the street. However, this raises the question of "what do the other players see"?
What if all the players - and even the NPCs - could rewind time, but they all saw it rewinding? So, if you get hit by a car, you rewind time. The guy driving the car then drives by with an apologetic wave: "sorry for willen-haven hitted you with my car!" He also witnessed the rewind, either as a player or an NPC.
Ignoring the obvious "how can I do my murdering" issues, let's look at a bigger issue. Now we run into the trouble that, in any population larger than maybe 10, we're going to be rewinding all the time, in very irritating ways.
There are several solutions to this, but for this particular post I've decided to localize the effects. The car hits you, and you rewind 10 seconds. Let's say you rewind very fast: 10 seconds in one "witnessed second". You and the car pass amicably after that.
The man on the street corner doesn't see ten seconds rewind. In his second of rewind, he only sees two seconds rewind. The man in his fifth-story apartment only sees half a second of rewind in that second. And the man watching a movie in the nearby cinema sees time slow to half for that second.
This gives us a very weird and interesting game mechanic. I recommend thinking about it for a bit before continuing, because while I have some ideas I'll show, it's a big enough thing that there are probably loads of ways to get fun dynamics out of it.
Lets put aside the various physicsical stuff such as shear and reversed light. Lets put aside the sociological ramifications of a world without accidents.
Instead, let's think about killing.
Imagine a typical multiplayer "fragfest" with these mechanics in mind. How would it work? Well, if you shot someone, they'd rewind time and get un-shot.
However, assuming you're beyond their "event horizon" (where their field goes from rewinding time to slowing time) you can continue to move forward while they move backwards, and continue to shoot them even as they rewind time. This doesn't help: your bullets will slow as they approach the event horizon, so you'll never hit them.
We can also assume that you might "fast forward" time in an attempt to create a conflicting field. But it would, at best, push the event horizon back towards the target, never actually letting you hit them.
So, how would you get kills?
Well, you could kill them such that they couldn't rewind time. If we presume they need brains to rewind time, headshots are a good way to do that. But I don't like that mechanic, so let's say that they can rewind time even after death.
You could kill them while they're in a state that they can't rewind time, such as killing them in their sleep without waking them up. But that's not a very common situation in a video game, and it's really sleazy besides.
But you could simply run them out of rewind, either because rewind is limited or because you run them back all the way to their spawn point and they "un-spawn". In the latter case, it's probably not a point: they probably just re-spawn somewhere else. The former case is fun and interesting, so let's put a cap on rewind capability: say, fifteen seconds of rewind at a maximum of two seconds per second (thirty seconds of history, max). That'll give us a fun chase scene.
There's also the idea of killing them because they're rewinding. For example, what if they crossed a bridge, and we break the bridge, throwing it into the ravine. During their rewind, they'll un-cross the bridge. The bridge bits are outside their rewind field, so they'll basically walk backwards and fall to their doom. This is assuming that physics apply in that way: we could also argue that they would walk backwards across "nothing" in the exact same manner they walked forwards, but I don't like that idea, because this way is much more interesting.
Simply erecting a spike wall behind them wouldn't work, because the wall would begin to rewind as the player rewound near it, and it would roll out of the way in plenty of time, back towards wherever it came from. Instead, the core thing would be to make sure that something that was there isn't there.
The game becomes a tactical game where you need to either chase someone back through time long enough to kill them, or change where they ran such that their rewinding screws them over, or both. For example, create a small pit using a grenade behind them. As they rewind, they get stuck in the pit, giving you an easy time chasing them down because they're not moving.
There are a lot of deep things we could do with this. Skill-wise, players learn to either hide their past or move to maximize safety. For example, a random jump now and then will offer some protection against the kind of trap I set in the previous paragraph, even though it made no sense at the time you originally did it. It becomes common to drop from heights, so that when you rewind, you scale a cliff that your pursuer can't follow you up. Mashers and stampers are an important element of safety: difficult to go through moving forwards in time, but guaranteed safe once you've gotten through and want to move backwards again. Not so for your pursuer.
You can set-up counter-traps: lay down a turret or a friend, move across its line of fire. Later, when an enemy is chasing you through time, they step right into the line of fire and have to rewind themselves. Because they've probably been "fast-forwarding" to keep your field small and to help keep up with you, they're probably close to out of juice and are easy prey.
Also, there are some things we can do by making certain exceptions for time rules. For example, let's say players are also telekinetic. While something is being telekinetically affected, let's say it is always aligned with your time field no matter where it is. To keep this sane, let's say you can only telekinetically pull, not push.
Telekinetically grab that can from over there! Woo! Okay, drop it, who cares. But when you're rewinding, that becomes a 2x-speed "bullet" fired from your rewinding self. If you've angled it right, that could easily slam into a pursuer for serious harm, especially if he's also moving at double speed "forward" to chase you.
If a rewinding enemy passes between you and a saw blade, pull the saw blade. It cuts right through the rewinding, so keep your eyes open and if you're worried about whether an object might be used to kill you, grab it and put it somewhere safe.
Out of these rules we've created an extremely distinct (if completely imaginary) shooter game where everyone can rewind time... but it's still a viable shooter and a very unique one.
There are loads of other things you can do with this "localized rewind/fast forward", I've just scratched the surface. What would you make?