I'll start this with a caveat: I tend to agree with the other answers that rolling for the sake of rolling should generally be avoided; that either narrating or letting the players narrate are generally better options.
Assuming you want rolls
Assuming you want the players to roll to perform routine maintenance:
- roll once per day (or week/month, depending on the frequency of encounters and length of average trips) per character
- let each character use a relevant skill (eg., the mechanic would roll an engineering check, the pilot would roll a navigation check, etc.)
- characters who don't need to maintain the ship (eg., Jayne or River) can roll a relevant skill to help with morale or maintain (or even train) in skills
- the results of the checks affect the state of the ship the next time it's relevant
Some ways that maintenance rolls might affect the ship and the journey (note: I've never played/read Traveller, so I'm assuming that there's some concept of a crit fail/crit success:
- failing a pilot check adds a fraction of the trip length (if you roll weekly, each failed roll adds a day; daily rolls might add an hour)
- critically failing a pilot check brings you into range of enemy sensors: evade or fight!
- success means normal progress is made
- critical success results in shaving some time off of the trip (or, if the pilot prefers, being better able to time their arrival; "we want to get there Tuesday afternoon" becomes "we want to show up Tuesday at 3:12 pm, from exactly this vector")
- failing a mechanic check adds time to the journey
- critically failing a mechanic check might reduce the maneuverability of the ship for a few days while the engines aren't reacting quite right
- success means normal progress is made
- critical success increases the ship's maneuverability until the engines get damaged
- crit fail: everyone has a (fairly minor; D&D equivalent of -1) penalty on all skills/attacks for the next time period
- fail: everyone has a (fairly minor; D&D equivalent of -1) penalty on ship-board maintenance tasks for the next time period
- success: status quo
- crit success: minor (D&D +1) bonus on everything for the next time period
... or, "Jayne plays with his guns"
- crit failure means that the gun monkey's favorite is unavailable for the next time period
- failure means that the favorite gun is operating at less than full potential (eg., -1 penalty) for the next time period
- success is a success
- crit success gives the favorite gun a bonus for the next combat (it's in peak working condition)
- I'm assuming that it's in the player's favor to have a +1 on a roll; switch signs if you're in a "roll under" system.
- I'm assuming that a bonus/penalty of 1 is meaningful (ie., that it maps relatively well to a 1 bonus/penalty in Pathfinder or D&D 3.5).
- If the system requires skill usage to improve skills, this is a good chance to get some low-risk skill usages under the player's belt.
- If a player has per-day abilities that affect particular skill rolls, let them use that ability on this check; if they're rolling less often than daily, they still only get to use it once.
thanby's comment reminded me of another option that I'd used in a Call of Cthulu campaign: dice pools.
In the CoC campaign, we had to stop a cult from summoning an Elder One (as per usual). As we succeeded at things, we added dice to a cup; when we failed, we added dice to the cult's cup. Right before the climactic battle, the cups rolled off: the power of our Elder Sign vs. the strength of their summoning.
Something like that could work, though figuring out who the "they" is might be tricky. So, just keep one cup of dice:
- crit success: add 2 dice
- success: add 1 die
- failure: remove 1 die
- crit fail: remove 2 dice
... then, use the dice either to augment rolls (Wash, landing Serenity near the end of the movie, used a lot of dice to augment his Pilot check) or for other bonuses (eg., "you got here faster than expected; you can get a 2% bonus for each of up to 5 dice").