One possible plot could be that the Joker kidnaps the girl to use her as a bargaining chip against Batman. He might think that Batman has a soft spot for this girl and would do anything to save her. So, he holds her in a secret location filled with his signature traps and chaos.
In a 'Joker kidnaps girl' fanfiction, perhaps the girl is a journalist who was getting too close to uncovering the Joker's latest evil plan. So, the Joker kidnaps her to stop her from revealing his secrets. He then taunts the police and Batman through messages sent with the girl's belongings.
In one scenario, the Joker might kidnap Robin to taunt Batman. He could hold Robin in a secret, elaborately booby - trapped lair. Robin, being the resourceful sidekick, would try to find ways to escape and leave clues for Batman. The Joker would then play mind games with both Robin and Batman through recorded messages and false leads.
Perhaps the girl outsmarts the Joker on her own. She uses her wits to escape from the place where she was held captive and then leads the police to the Joker's hideout. So, the fanfiction ends with the girl being a hero herself.
The Joker might try to make Batman laugh at his own jokes as a form of humiliation. He could use his trademark laughing gas in a way that forces Batman to either laugh or suffocate.
In this fanfiction, it could be that Loki was misled into thinking the girl was an enemy spy. So, he kidnaps her to interrogate her. However, during the interrogation, he discovers she's completely innocent. Now, he has to figure out how to make up for his mistake without losing his pride, and also deal with the potential chaos that his hasty action has caused among her friends and family.
Nightwing could also be a likely savior. He was once Robin himself and has a strong bond with the current Robin. He might take it upon himself to rescue Robin, using his own unique fighting style and experience to outwit the Joker. Plus, he has his own network of contacts in Gotham that could help in the search.
One idea could be that the Joker kidnaps Batman to force him into a series of cruel 'games'. Maybe he wants Batman to choose between saving his closest allies or innocent civilians, testing his moral limits.
One possibility is that Artemis has been misinformed. Some enemy of Percy has tricked her into thinking that Percy is a danger to the demigods. So she kidnaps him to stop him from causing harm. But as she keeps Percy captive, she slowly realizes that she has been deceived when Percy starts to explain his side of the story and the real threats he has been fighting against.
The plot could involve the Phantom seeing Christine as his only salvation from his lonely and tortured existence. So, he kidnaps her with the intention of making her understand his world. He may show her his musical creations, hoping she'll come to love him in return. However, Christine might be initially terrified and try to escape.
In some fanfictions, Aizawa might be under the influence of a villain's quirk that makes him act against his will. So, he kidnaps Izuku, but later on, he realizes what he's done and tries to make amends. This could lead to a story full of drama and redemption.
One possibility could be that Minato was under some kind of mind - control spell. Maybe an enemy of Konoha managed to manipulate him into thinking that Naruto was in danger by being in the village. So, Minato kidnaps Naruto to take him to what he believes is a safer place. But as the story unfolds, Naruto tries to figure out what's wrong with his father and tries to break the spell on him.