It could be a former criminal that JJ put in prison and who has now been released and seeking revenge. Maybe it's someone from her past that she had a run - in with during her early days in the BAU.
There's also the possibility that the unsub is a relative of a previous victim that the BAU, with JJ's help, failed to save. This person has been consumed with grief and anger and has decided to take JJ as a way to make the team feel the same pain he has felt all this time.
Perhaps it's a person who has an obsession with the BAU in general and sees JJ as the most vulnerable or the most important link in the team. It could be a disgruntled former law enforcement officer who blames JJ for ruining his career and now wants to make her pay. He would be smart and know how to cover his tracks, making it difficult for the team to find her.
One possible plotline could be that the unsub kidnaps JJ because she has been getting too close to uncovering his identity in a series of murders. The team then has to race against time to find her. They might start by retracing her last steps and interviewing people she was in contact with recently.
In the fanfiction, JJ gets kidnapped right outside the BAU building. It's a bold move by the kidnapper. The team is stunned. Emily is the first to start piecing together possible motives. They discover that the kidnapper has a vendetta against the BAU. JJ, while kidnapped, manages to leave some subtle signs for the team to find. The team races against time, using their combined skills of profiling, forensics, and technology to track her down before it's too late.
One possible motive could be revenge. Maybe JJ helped put away a family member or someone close to the kidnapper, and now they want to make the BAU suffer by taking her. Another could be that the kidnapper is a serial offender who targets strong, intelligent women like JJ as a way to assert power and dominance.
It could be a disgruntled former crew member. Maybe they were fired from the show and in their anger, they read a lot of fanfiction for inspiration on how to get back at the team. So they plan and execute the kidnapping.
One possible motive could be revenge. Maybe Reid has put away a relative or someone close to the kidnapper in the past. Another motive might be related to Reid's intelligence. The kidnapper could want to use his brainpower for some illegal or self - serving scheme, like cracking a code or coming up with a complex plan.
It could be someone Reid put away in the past. Maybe a criminal who has held a grudge against him ever since and has now found an opportunity to kidnap him for revenge.
Hotch would immediately take charge and start organizing the team. He would be calm but determined to get JJ back. Morgan would be on the front line, ready to use his physical skills to track down the kidnapper. He'd be really angry and motivated to rescue JJ as she's like family to him.
There could be. 'Criminal Minds' has a large and dedicated fan base, and fanfiction is a popular way for fans to expand on the relationships in the show. Some fans might be interested in seeing a romantic moment between JJ and Reid, so it's possible that such fanfictions exist. You can try searching on fanfiction websites to find them.
Well, perhaps the kidnapper is a copycat of a previous unsub that JJ helped put away. He wants to make her suffer as he believes she ruined the original unsub's 'legacy'. So he takes her to a remote location, and while she's trapped, she tries to use her profiling skills to understand his motives and find a way to escape or signal for help.
It could be a deranged fan of one of Reid's academic works. This person might have misinterpreted Reid's theories and decided that Reid was some sort of false prophet. So, in his twisted mind, he kidnaps Reid to 'correct' him.