Well, in a 'pit gets kidnapped' fanfiction, the plot could involve the search for Pit. Maybe his friends like Palutena start a quest to find him. They might face various obstacles and enemies along the way.
It could be Hades. He's always been an antagonist in the series and might kidnap Pit to mess with the gods. He could be thinking that by taking Pit, he can gain an advantage in his ongoing feud with them.
I'm not sure which particular fanfiction you're referring to. Some fanfictions do involve Pit getting kidnapped, but without knowing the specific one, it's hard to say for certain.
One idea could be that Pit is kidnapped and Hades tries to corrupt him. Maybe Hades offers Pit power in exchange for his loyalty. Pit, being the heroic character, resists but has to find a way to escape while also facing the internal struggle of being tempted by such power.
In many fanfictions where Gil gets kidnapped, it could be for various reasons. Maybe it's by a rival group who wants to use him as a bargaining chip against his friends. For example, in some stories, Uma's crew might be targeted because they are seen as a threat in the Isle of the Lost. Gil, being a part of that crew, gets kidnapped. Writers often use this plot device to create drama and show the loyalty and courage of the other characters as they try to rescue him.
Well, Hades might try to force Pit into doing his bidding. For example, he could make Pit fight against his own friends. But Pit, being a strong - willed character, would likely try to find a way out of this situation. He might start by trying to find a way to break free from whatever bindings Hades has put on him.
One idea could be that Sam is kidnapped by a group of rogue hunters who think he's gone too far in his use of demon powers. They lock him in an abandoned bunker and try to 'exorcise' the 'demon' out of him, not realizing he's in control. His brother Dean has to race against time to find him, following a series of cryptic clues left by the kidnappers.
Natsu was walking through the forest one day when suddenly a group of dark mages emerged from the shadows. They quickly overpowered him with some sort of magic -binding spell. They dragged him away to an old, creepy lair. His friends back at the guild soon realized he was missing and set out on a search mission. Happy flew around frantically looking for any sign of his friend. Lucy, Gray and Erza teamed up and followed some leads they found in the forest. They were determined to rescue Natsu no matter what it took.
Peter's kidnapping was a carefully planned event. The kidnappers had been observing him for weeks. One day, as he entered a park, they grabbed him. They blindfolded him and took him to a secluded location. Peter was scared but he didn't show it. He started to befriend one of the kidnappers, a young guy who seemed a bit hesitant about the whole thing. Peter told him stories about his life and gradually the kidnapper started to have second thoughts. Eventually, the kidnapper helped Peter escape by distracting the others.