Another plot could be that Kayaba has some sort of vendetta against Kirito. Maybe he sees Kirito as a threat to his own power and control within the virtual world. So, by kidnapping him, he aims to either break Kirito's spirit or get him out of the way so that he can maintain his dominance.
It could be Klein. He is Kirito's friend and has his own set of skills. He might gather a group of other players who are also concerned about Kirito's disappearance and together they try to rescue him from Kayaba.
One common plot could be a super - fan being kidnapped by someone who wants to use the fan's knowledge of the Beatles to find some hidden treasures related to the band, like unreleased songs or valuable memorabilia.
In some fanfiction of this kind, the monkeys might kidnap a human because they think the human is a key to unlocking a hidden treasure or ancient knowledge. They see the human as having some unique quality like intelligence or intuition that they lack. So they take the human to their secret lair in the jungle and try to communicate with them to get what they want.
One interesting Naruto Kayaba fanfiction could be about Kayaba somehow being transported to the Naruto world. Maybe he uses his knowledge of technology to create new jutsu or devices. For example, he could combine his virtual reality skills with ninjutsu to create training simulations that are far more advanced than what the ninja in Naruto are used to.
One possible reason could be for some sort of experiment. Gaster is often depicted as a mysterious scientist in the Undertale lore, and Sans has unique abilities. Maybe Gaster wants to study Sans' power over time and space. Another reason could be related to secrets. Sans might know something about Gaster's past or the true nature of the Underground that Gaster wants to keep hidden, so he kidnaps Sans to prevent him from revealing it.
Another motive could be related to some sort of power play. Eric might want to use Sookie as a bargaining chip. Since Sookie has her special abilities, other powerful entities in the fanfiction world might be interested in her. Eric could kidnap her to gain an advantage over his rivals.
One reason could be that Snape believes Hermione has knowledge that is dangerous if it falls into the wrong hands. So, he kidnaps her to keep her safe until he can figure out what to do with that knowledge. Another possibility is that he is being controlled by some dark magic or an outside force which compels him to take her against his will.
In some fanfiction, it could be due to a prophecy. Percy might think that by kidnapping Annabeth, he is fulfilling a part of a prophecy that would ultimately lead to a greater good. He could be trying to keep her away from a certain situation or enemy that the prophecy foretold. However, his way of handling it, which is kidnapping, is a very extreme measure.
One possible motive could be that in the fanfiction's plot, Illya might be influenced by some dark magic or external force that compels her to kidnap Shirou. For example, if there is an evil entity controlling her actions, it could use her as a tool to get to Shirou for some unknown, perhaps sinister purpose related to his powers or his connection to other characters in the story.
It could also be for power. In the fanfiction world, Sebastian might think that by kidnapping Kurt, he can gain some sort of power over the New Directions or the McKinley High glee club. He might want to use Kurt as a bargaining chip to get what he wants from them.