Perhaps a deranged fan of Beckett. Maybe Beckett is a well - known figure, like an actor or a writer, and a crazy fan who has an unhealthy obsession abducts them. The fan could be thinking that by doing this, they can get closer to Beckett in some warped way.
It could be about a father - son duo named Beckett who are suddenly kidnapped. Maybe they are abducted by some criminals who want to use them for ransom. Or perhaps they are abducted by a mysterious group with a hidden agenda related to their family secrets.
One possible plot could be Beckett's intense search for the abducted child. She might use her detective skills to follow leads, question suspects, and piece together the mystery of the abduction. Another plot could involve the emotional journey of the people related to the abducted child, like the parents, and how Beckett interacts with them during the investigation.
It could be a criminal with a vendetta against Beckett. They might abduct the child to get back at her for putting their associates in jail in the past. This antagonist would be calculating and always one step ahead, making it a real challenge for Beckett to rescue the child.
One common theme is family bonds. Since it's about a father - son relationship, many stories explore how they communicate, misunderstand, and then reconcile. For example, a story might show the son struggling to live up to the father's expectations at first, but then they grow closer as they understand each other better.
It could be a criminal mastermind who has a vendetta against Beckett or someone close to her. This antagonist might see the baby as a way to get back at Beckett, either by using the baby as bait or harming it to cause her pain.
One possible plot could be that Beckett embarks on a dangerous journey to find the abducted baby. Along the way, she uncovers a complex web of secrets and has to outwit the kidnappers. Maybe she teams up with an unexpected ally who has some inside knowledge about the abduction.
A great fanfic usually has well - developed characters. The son should have a distinct personality. Also, the relationship between Castle and Beckett as parents should feel real.
I also suggest checking out some fan - run forums dedicated to the relevant fandom. Fans there might share their own written Beckett and Son fanfic or point you to where you can find the best ones. They could also give you some insights into what makes a really good Beckett and Son story, such as strong character development and an engaging plot.
You can start by looking on fanfiction websites like Archive of Our Own (AO3). There are often tags and search filters that can help you narrow down to Castle and Beckett stories involving their son. Another option is FanFiction.net, which has a large collection as well.