It could be Nightwing. He was once Robin himself and has a special connection to the current Robin. Nightwing might hear about the kidnapping and step in to save Robin, using his own unique fighting style and experience.
One possible plot for this fanfiction could be that Bane kidnaps Robin as part of a revenge plot against Batman. Robin is trapped in a dangerous situation, and he has to rely on his training and his mental strength to survive. Bane might taunt Batman with messages about Robin's kidnapping. As Batman races to save Robin, Robin also tries to find a way to free himself and help Batman in the process. The story could explore Robin's growth as a character while facing this difficult situation and the complex relationship between Batman, Robin, and Bane.
Robin would probably be both scared and angry. He's a brave kid, but facing the Joker alone is no small feat. He might initially try to fight back if he has the chance.
The other Teen Titans could also play a big role in the rescue. Starfire could use her powers to fly in and distract Slade while Cyborg and Beast Boy create a diversion. Raven could use her magic to break any spells or traps that Slade has set for Robin. They would all work together as a team to get Robin back safely.
Robin could use his intelligence to analyze Deathstroke's lair and find a weakness in its security system. He might notice a pattern in the guards' patrols or a flaw in the locking mechanisms. Then, he could use his acrobatic skills to create a distraction and slip away unnoticed.
Robin would probably stay calm and observant. He's been trained well, so he'd immediately start looking for any weaknesses in his captivity situation. He might try to pick the locks on his restraints if he's bound, or find a way to communicate with his team secretly if there's a chance.
A more unorthodox plan could be for Oracle to spread false information about a powerful new weapon that only Robin knows how to operate. This would make Luthor believe that Robin is more valuable alive than dead. Then, while Luthor is distracted trying to figure out how to use Robin to get this weapon, the heroes could stage a rescue. They could also try to turn one of Luthor's minions against him. Offer the minion a deal in exchange for information on Robin's whereabouts.
Robin might initially be defiant. He's a trained fighter and detective, so he'd likely try to gather as much information as he can about his surroundings. He could be looking for any weaknesses in Lex Luthor's security setup or trying to figure out what Luthor's endgame is.
One idea could be that Poison Ivy kidnaps Robin to use him as a bargaining chip against Batman. Maybe she wants Batman to stop interfering in her environmental plans. So she holds Robin in her lair, which is filled with all kinds of dangerous plants.
In the world of fanfiction, there are often hidden gems. For 'Robin kidnapped' fanfictions, you might find good writers by looking at the reviews and ratings of stories with this plot. Writers who get a lot of positive feedback for their take on this theme are likely to be good. They may be skilled at character development, making the story of Robin's kidnapping and possible rescue really engaging.
It could be the Joker. He's always up to no good and kidnapping Robin would be a great way to mess with Batman. The Joker might want to use Robin to force Batman into a series of cruel and twisted games.