Couple random ideas - I don't think they've been done directly before.
"Between a Rock and a Hard Place"
A sentient species of cave-dwellers is currently having a problem. They live on a harsh environmental world in the Klingon-Federation frontier and are seriously agoraphobic and so never developed space flight or warp travel. Neutrino and subspace emissions indicate advanced technology, but they may not even conceptually be able to deal with 'outer space', so the Federation didn't contact them, the Klingons didn't when they were friends with the Federation and then were too busy fighting the Federation to set up an occupation force.
That said - a hefty extinction-level asteroid is headed their way and will impact in a few months, and with things calming down in the Neutral Zone, your captain on patrol picked up the upcoming problem on an astrometric survey, that the species itself is slowly becoming aware of.
Socially, they're an insular culture, and so were left out of contact for fear that it would destroy them culturally from the Federation perspective. Thanks to previous peace treaties, the Empire hasn't managed to exploit them quite the way they'd like, so they're experience with both powers is a blank slate.
What does your captain do? Try a temporary transplant? Reveal yourself as the terrifying 'Sky God' and swear their loyalty to your House in a fairly bloodless takeover? Reveal yourself as the terrifying 'Sky God' and let the rioting run its course to rebuild the culture into a more 'normative' form to induct them into the superpower of your captain's choice? Infilitrate and try to put together an interception program in a few months? Or just replicate a bag of marshmallows, let nature take its course, report on the sadly empty planet, and see if you can get ten percent on the proceeds for turning the place over to a horde of hungry Horta?
"I will wait forever"
With the Preservers still busy integrating with the Deferi, more and more are being awakened and ranging farther afield all the time as tourists and storytellers par excellence. Your captain or one of your bridge officers/crew encounters one of these travelers, and find to their surprise/amusement/disgust that the Preserver is convinced you are a reincarnation of a lost lover or family member of their own genetic history line. It may not be a major event, but certainly should be worth a round of drinks for the retelling before it's over.
"The Rise and Fall of the Delta Quadrant"
It's been noted before that each Quadrant seems to have its own governing style among the strongest players. With the Alpha Quadrant characterized by the republican Federation, the Beta by the imperial structures of the Romulan (until recently) and the Klingon Empires. The Gamma has the totalitarian theocracy of the Dominion....
and the Delta Quadrant seems defined by utter anarchy, with historically, strong governments going imperialistic and being pulled down by loose coalitions. Combined with the Quadrant's generally longer history of warp travel versus the expansion rate of the Alliance governments, and with the Borg more or less falling into this model at this point, it's starting to be noticed as a trend.
Somehow, your poor captain and his or her crew has been shanghaied into an archeological dig as a result on one of the oldest ex-capital worlds that friendlies in the Delta Quadrant have found in their histories. What's there to discover? Iconian interference? Ancient conspiracies? Or maybe the researchers just got the weight wrong when they replaced the idol, and would really like a beam-out please.
"Fate: Protects fools, little children, and ships named Enterprise." - William Riker