I like the idea of Star Trek more than Star Trek itself. Why? Probably because Star Wars exists and everything in it looks cooler. But anyway...
Here's a game Mandy and I thought up just before bed at the end of a long discussion about how RPGs work . It's called...
Star Trek Poker or, The Star Trek RPG That Actually Plays Like An Episode of Star Trek
(Probably also adaptable to simulate other hour-long space-travel shows like Galactica, Firefly, Babylon 5 etc, but I can't be sure because I never watch them for reasons I won't elaborate here because I don't want to risk further alienating my audience.)
Ok, so first off:
You generate characters as normal using whatever system. Strength, intelligence, weird abilities etc. You can use one of the 3 (?) systems that already exist for Star Trek RPGs or Savage Worlds or GURPS or whatever.
HOWEVER, in addition to all that the character will get a "position"--this is basically his or her "role" in the show--it can be a formal position on the ship or just what they tend to be in charge of. Positions can include: Tech (Geordie, Scotty, O'Brien etc.), Captain (Picard, Kirk, etc.), Security (Worf originally, Odo), Omnicompetent (Spock, Data, Dax, 7 of 9, etc.), Psychic (Troy), Doctor (Bones, Bashir, Crusher, etc.), Seemingly Useless 2nd Stringer (Nog, Barkely, etc.)
The "position" allows you a special "power" which you can use--under certain very specific circumstances--to completely change the plot in a Forgeite/narrativist sort of way.
The Tech person can solve any one engineering problem, the Captain can convince NPCs and foes to do things completely counter to their nature, the Security person can get previously unknown information out of nowhere, the Omnicompetent has no special powers but gets to have a bonuses to all of his or her ordinary stats--s/he gets more "build points" if the system uses that, the Psychic can reveal (make up) secret facts about NPCs, the Doctor can solve any biological challenge, and the Seemingly Useless 2nd Stringer can do any of these things but at a lower rate of success.
Which brings us to the mechanic for these powers...
Basically, the GM has to build a specific kinds of adventure--or "situation"--and this situation MUST be complicated by:
A technical obstacle, a dealing-with-unfriendly-aliens obstacle, and a biological obstacle.
For example: the ship is stuck in a [insert pseudoscience here] field while aliens that have the ability to replicate their own bodies using the ship's food replicators are invading the ship and chewing apart the guidance sensors.
In addition to all the work that needs to be done to make any RPG adventure, each of the three sub-problems must have a "difficulty level" on a scale of 1-10. The GM has 10 "difficulty points" to allocate. So, for example in this adventure the GM could decide that the technical obstacles are a level 6 problem, communicating with the aliens is a level 3 problem, and understanding the alien genetics is a level 1 problem. The GM writes all these numbers down and keeps them secret.
Also, there should be a "ticking time bomb" type-element to an adventure, where certain things, if they are allowed to occur, will have permanent negative consequences. Like if the aliens aren't stopped then the crew wil have to abandon the ship and their next adventure will take place in an escape pod.
Now, the PCs start each episode with zero "plot points". They go along playing the game like you would any RPG set on a spaceship--however whenever any player achieves something positive (shoots a hostile alien, succesfully pilots through an asteroid field, etc.) he or she is awarded a number of Plot Points.
The Plot Points are shared by the PCs as a whole. At any given time, rather than addressing the problem normally, the crew may decide to "bank" on a given PC's special power. (Usually after a sort of conference with all the PCs present.) That is--they tell the GM they are going to spend 6 Plot Points on, for example, the Tech guy's abilities. (The 2nd stringer can only use 3/4 of the available plot points--rounded down.)
At this point, the GM reveals the Difficulty Level of the relevant challenge. If the number of Plot Points "banked" on the power is greater than the Difficulty Level, then the crew "solves" that problem in some way--narrated by that player.
Like if the crew decides to bank 6 points on Geordi and he's facing a 5 point technical challenge, the person playing Geordi can announce "All I have to do is modulate the linear induction around the phase coils! Of course!". And that'll work. If the GM revealed that the technical challenge was actually an 8-point challenge, then it wouldn't work.
Either way, the crew then loses all the banked Plot Points (and precious time). If they fail, then that means they've got to either to do more stuff in the traditional way to solve the problem, or else build up more plot points to "gamble" again on a different aspect of the problem. Like if the technical problem turned out to be level 8, then that means that the diplomatic problem can be a maximum of level 2 so all they need is to get 2 more Plot Points before the ship blows up and they can save themselves by making peace with the aliens.