Perhaps Hinata is kidnapped because she has some unique power or knowledge that the kidnappers want. In the fanfiction, the story could focus on how she uses her wits and hidden strength to try to escape while her comrades search for her. For example, she might be able to communicate with a small animal in the cell she's held in and use it to send messages.
It could be a rogue ninja who has a grudge against Hinata's clan. Maybe this ninja was once rejected by the clan and now wants revenge by taking Hinata.
There's also a possibility that Hinata's own clan, the Hyuga, would take matters into their own hands. They have their own unique techniques and a sense of pride. They would not allow the Akatsuki to keep one of their own captive for long and would send their strongest members to retrieve Hinata.
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.
Hinata could use her Byakugan to find weaknesses in Sasuke's defenses. Maybe she spots a chakra - based trap that she can manipulate to create a distraction and then make her escape. She could also use her Gentle Fist techniques in a creative way, like redirecting Sasuke's attacks back at him to create an opening for her to run.
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.
It could be a jealous rival player from another school. This player might feel that Hinata is getting too much attention and success in volleyball, so they decide to kidnap him to stop his progress.
In a 'hinata shouyou kidnapped' fanfiction, the plot might involve Hinata being kidnapped by a mysterious figure who has a vendetta against his volleyball skills. This person wants to break his spirit. Hinata is initially scared but then realizes he can't give up. He starts to look for weaknesses in the kidnapper's hideout. Meanwhile, his coach and team notice his absence and suspect foul play. They follow some clues that Hinata cleverly leaves behind and are able to rescue him. After his rescue, Hinata comes back stronger than ever, determined to show the kidnapper that he can't be broken.