A few loose ideas: (disclaimer, might have been proposed by others somewhere...)
*Use specially assigned diving bell to salvage items from submerged wreckage. (Diving could be quest-specific item... no one lugs around a diving bell for fun methinks)
*Alternate: Protect ship that is recovering items from wreckage with diving bell.
→Diving bells would make some lovely but possibly intensive animating... too much extra work for such a small feature?
*Investigate and intervene in coastal town that has gone rogue. →Revolt in a town has led to them declaring independence from any existing nation.
→Could be a smaller ship mission, depending on town size. (This could also require some "diplomacy", and off-ship action)
*Similar scenario could be with natives overrunning a small coastal town or fishery.
*Siege assist mission: A city/fort/town is under siege from land. Rescue peeps while being under fire from coastal positions (outside of town), if any. Could elaborate on possibilities on this...
Introduce a few enemy ships in port to mix things up, have the town already partially taken over including some coastal batteries (could progress during mission... more fire to endure as mission progresses).
*Rumor-based missions: Something along the lines of sighting of a ghost ship. Big difference from other missions would be decreased morale for crew and officers, in fear of the supernatural.
→I do NOT advocate supernatural stuff or magic in the game, but am only drawing on the intense superstition that was common in that era. Final target could be smuggler trying to look scary but 'maimable' flesh and blood all the same.
Not sure if reputation will be in, but:
*Disease control: have a choice of either picking up and transporting quarantined populace, buy and deliver supplies, or outright killing all infected.
→In a larger picture, I wouldn't mind having "choices" such as that in missions.
*Not really a mission per se, but how about having the choice between hiring at a tavern (higher cost, higher morale) or pressing crew into service (lower cost, lower moral, possible friction with locals)
(Could be a mission if ordered by the admiralty to increase crew to a minimum level without the option of "hirable" crew present: choose to ignore the admiralty or rake in some scum via impressment)
*Missionary mission: transport a missionary to a coastal area that has few to no modern ports: possible attacks by natives, where you have to weigh safety and diplomacy (killing everything in sight will leave few to convert for the missionary, plus they won't be as easily converted either), with unmapped coastal hazards thrown in.
(I can already see chat: "Missionary, anyone?")