What about a Wesley Crusher/Traveller series where the Traveller took Wesley as his companion on adventures through time and space?
What do you mean Doctor Who already did it?
Or Counsellor Troi could become an empathic criminal investigator. It would be just like Psych without the clever central conceit.
Or Geordi and Data could quit their jobs with Star Fleet to pursue their dream of being performers. Due to a misunderstanding they quickly find work as a ventriloquist duo, but what the audience doesn't realise is that Geordi isn't a ventriloquist. Misunderstandings and hilarity (fail to) ensue.
Okay, they might be a bit high budget so how about several centuries in the future Data tells his children about how he met their mother... really just a framing device for repeats of TNG episodes.