Well, sometimes in fanfiction, Red Robin is kidnapped for ransom. Maybe some small - time crooks think they can get a huge payday from Batman or the other heroes. They don't realize how much trouble they're in for. Another idea is that a magic - using villain kidnaps Red Robin because they sense his great intellect and want to use his mind to figure out some ancient magical riddle. It's a different take on the typical kidnapping scenario.
A common plot is when a deranged fan kidnaps Red Robin. This person might be obsessed with him and wants to keep him all to themselves. They could have a warped view of what it means to be a 'fan' and think that by kidnapping him, they can have a special relationship with him. In some fanfictions, this leads to Red Robin having to use his wits to escape while also trying to deal with the psychological aspect of being kidnapped by someone who claims to admire him.
A common plot is the revenge kidnapping. Maybe a former foe of Batman targets Robin to get back at the Dark Knight. For example, a criminal whose operation was shut down by Batman decides to kidnap Robin to draw Batman out and make him suffer.
One possible plot could be that Robin is kidnapped by a new villain who wants to use his knowledge of the Young Justice team against them. The team then has to race against time to find him and stop the villain's evil plan.
Another plot could involve Sakura being kidnapped by an enemy who wants to use her for her special abilities. This could lead to a story of her trying to escape on her own while also dealing with the kidnapper's demands and threats.
A possible plot is that a group of rogue ninjas kidnaps Sakura to lure Sasuke into a trap. They might want to steal his Sharingan or get revenge on him for past deeds. Sasuke, being the skilled ninja he is, uses his intelligence and combat prowess to outwit the kidnappers. He may also get help from his old friends like Naruto, but the focus remains on his relationship with Sakura and his determination to bring her back safely.
Another plot could involve Batgirl using her wits while kidnapped. She might try to escape on her own by tricking the kidnappers or finding a weakness in their security. For example, she could pretend to be sick to get closer to a guard and then overpower them.
Villains from the Batman rogues' gallery are likely kidnappers. For instance, the Riddler might kidnap Red Robin to challenge Batman to solve his riddles to get him back. Another one could be the Penguin. He might see Red Robin as a valuable hostage to bargain with for more power or money in Gotham's underworld. Black Mask is also a possibility. He's always looking for ways to gain an edge over the other criminals and having Red Robin as a captive could be part of his scheme.
Another plot might involve Star being kidnapped by an enemy who wants to use her magic for their own evil purposes. For example, a dark wizard could hold her captive and try to extract her magic, while Star has to use her wits to either escape or hold off the wizard until help arrives.
One common plot could be the rescue mission. The SVU team races against time to find Olivia. They follow clues, interrogate suspects related to her kidnapping. Another plot might involve Olivia's inner strength during captivity. She could be trying to outwit her kidnappers, leaving behind small signs for her team to find. Or perhaps there's a plot where the kidnapper has a personal vendetta against Olivia due to a past case she worked on, and the story delves into the backstory of that relationship.
A common plot in 'Aria Gets Kidnapped Fanfiction' is the psychological battle. The kidnapper may try to break Aria's spirit, but she refuses to give in. She uses her mental strength to outwit the kidnapper. For instance, she might pretend to be compliant while actually looking for ways to signal for help. Another aspect could be the discovery of new powers or abilities within Aria during her captivity. This could be something like she discovers she can communicate telepathically with a nearby animal who then helps her escape. Also, there could be a plot where the kidnapper is actually a misunderstood character, and through her time with him, Aria comes to understand his motives and tries to reform him.
One common plot could be that Amy Rose is kidnapped by a villain who wants to use her as bait to lure Sonic. The kidnapper might think that Sonic would come to save her, and they could set up a trap for him. Another plot could be that the kidnapper has a personal vendetta against Amy Rose herself, perhaps because she got in the way of their evil plans before. There could also be a plot where Amy Rose is kidnapped by mistake, and the kidnapper realizes their error but still decides to hold her captive for some other reason, like ransom or information.