Siobhan and I have done a little brainstorming work for a new roleplaying campaign we might be starting up. The idea is one of those that's been on my ideas pile for a while, but she will probably end up GMing it.
The central premise is a different approach to time travel. We first assume that everyone is reincarnated over and over, life after life. Then some people come up with a technique, not to travel back in time, but to transfer their consciousnesses into their previous lives. This lets us have adventures in many periods with a different twist from most time travel, and without some of the pitfalls and paradoxes of time travel. For instance, if things go wrong, you can't just go back to a few days earlier and fix them.
If you only had a single player character, this would be almost all you needed, but to run this with a group of players you need another element. If in a previous life I was an astrologer in the court of Charlemagne but at the same time you were a farmer in the Scottish highlands, it'd be awfully hard for us to share an adventure. So we further posit that we all travel in circles of "kindred souls": the people who are around us now were, in previous lives, also around us, because souls form a connection and find each other. This gives us some interesting conclusions. Anyone who's always felt like they didn't fit in probably felt that way because they weren't near their kindred souls yet and still needed to find them. Sometimes when we meet someone we immediately like or hate them without any reason: that's because of souls being kindred (or not). It even makes it possible to set up a recurring nemesis if desired.
Displacing your past self also gives players a chance to have some interesting character creation fun, though at the cost of forcing us to use a very simple system where character creation is simple and quick, since an abbreviated version of it will have to be done each adventure. In essence, every adventure, you get to make a new variation on your character. First, figure out what your life was in the time and place specified; second, figure out what results from the combination of your current self and past self, when the consciousness of one occupies the other. You'll remember enough of both lives to be able to navigate the time period and life you're in and still remember why you're there. If the player wants to be very consistent, they can be the kind of soul that ends up in a very similar body, with a very similar demeanor, every time, and even avoid the character retooling almost completely. But they can also be the kind of soul that changes a lot; the body can change size, race, and gender, the personality and skills can change completely. Either end or anything in between is completely justifiable. Personally, I'd likely go the latter, as I like the idea of a new variation on my character every adventure.
Most time travel campaigns are based on the idea that the characters have at least some control over the time travel process, but this gets you into all kinds of trouble: why not just go back to a few days earlier and fix whatever went wrong, for instance. It also means everyone wants to go kill Hitler, save JFK, talk to Jesus, slap a tracker on Amelia Earhart's airplane, watch the world premiere of Romeo and Juliet, etc. These are all situations prone to being tricky to GM and many of them are kind of worn out. Then there's all the "save the world" things, where you're trying to stop aliens or people trying to change the timestream, which also are overdone and invite some problems.
Instead of doing that, we're going with an archaeology/academic approach. The characters will be trying to retrieve interesting artifacts (but not in the Warehouse 13 sense -- not superpowered artifacts, just historically interesting ones). This gives the GM a good pretext to get the characters to go on the adventure she's prepared ("Your research has revealed that the long-lost Sceptre of Queen Neferhotep was seen in Barcelona in the late 1580s, and your past lives at that time were not far away, in Basque country.") rather than deciding to wander off on their own tangents. However, it does mean that every adventure either has to be a heist or dungeon crawl (since its objective is to get an item), or we have to use the Scooby Doo approach, where going after whatever we're going after always happens, by amazing coincidence, to involve us in whatever interesting thing is happening in that time). We are still seeing if we can retool this to avoid the coincidence but still give us a reason to participate in interesting historical incidents.
Of course if only your consciousness goes to the past you can't bring an object back to the future with you. If we change the pretext to where the characters are trying to gather information instead of objects, that issue disappears. But if we stick with objects, it leads to another interesting element to each adventure. Having secured the item, what next? Maybe you bury it in a cave that you're fairly sure no one will find in the ensuing centuries, then when you wake back up in the present, you go dig it up and hope it wasn't found. Maybe you rent a safe deposit box. Maybe you use your extensive knowledge of archaeology to identify a tomb in the past that no one has discovered by the present, discover it, break into it, and hide the item there, then go find it in the present. In fact, getting into the safe enough place to hide the item might be more of an adventure than getting the item.
In addition to working out the mechanical details about how characters are created and merged, what happens if they die, what limitations there are on time travel, etc., we still need to decide if time is elastic. Can you change the past, or do your activities there end up explaining the record as it's already known? Consider this in the case of that sceptre I mentioned before. If time is inelastic, and whatever happened, happened, then they can simply check before they leave to see if that Sceptre was ever seen at a later date. If so, they won't bother to go to Barcelona since it's clear they won't succeed in getting it. This means the only trips they'll take are to the last moment the item was seen (or after that), which requires more coincidences -- now the last moment has to be one where the characters have a previous life nearby.
Or is it a coincidence? Perhaps the reason it's the last sighting is because that past life allows the characters to get it from then and hide it, thus ensuring it won't be spotted thereafter. In some ways, inelastic time fixes some problems with time travel and makes for a far more interesting story, but it has some problems for the GM: the characters can try to research their past lives enough to find out what happened, and once they know, that has to end up happening, which can be tricky for the GM to arrange without railroading. Cooperative players can greatly minimize the difficulty here; they just have to try not to contradict the known timeline, perhaps because if they control the situation which leads to a known outcome, they can control what it really means. ("It only seemed like Feliz died, but we faked his death and arranged a new life for him." Trying to avoid his death would have failed, so why bother? Instead, own it and take it over.) The GM can also get a lot of mileage out of the fact that the details of the lives of unimportant individuals are generally not recorded at all in the past.
An elastic past is probably easier to start with, but it also tends to create the big problem that you have to figure out what effect the changes in the past will have on the present. The "course correcting universe" thing feels like a cop-out to me (where most changes you make in the past get cancelled out and everything happens to turn out status quo), but anything else can cause campaign explosion by butterfly effect, or forces the characters in the past to have to tread on eggs everywhere they go (this challenge is interesting once in a while but having to sustain it tends to eliminate a lot of interesting storylines and lead to a lot of intentionally dull stuff). Usually if you have an elastic past, no matter what you try to make the game about, it ends up being about changing history, or avoiding doing so.
Once those issues are addressed, the premise has the opportunity to let us play around in a lot of history visiting interesting times and places, without running into the paradoxes and pitfalls of most time travel stories, or the overdone themes. Now we just need a good catchy name.