Well, it could be that Beckett's mystery illness is actually a curse. In the fanfic, there might be a magical or supernatural element. For example, an old family feud with a witch or a wizard who put a curse on Beckett, and the story could revolve around breaking the curse to cure the illness.
Beckett is obviously one of the main characters as the one with the mystery illness. There might also be a doctor character who is determined to find out the cause of the illness and treat it.
One possible plot could be that Beckett is kidnapped by a rival group who wants to use his skills or knowledge for their own gain. His friends then embark on a dangerous rescue mission.
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 also be a fanfic that takes place in an alternate universe. In this universe, the events of 4x23 didn't happen as they did in the show. Maybe Beckett didn't become a detective in the same way, and Castle meets her in a completely different situation. They then start to solve mysteries together in this new setting.
One possible plotline could be that the baby has some sort of special ability or is in some way connected to a mystery they have to solve. Maybe the baby's birth coincides with a strange series of events in the city, and Castle and Beckett have to use their skills to protect the baby and figure out what's going on.
One possible theme could be friendship. As they interact in the hotel environment, they could form a deep and meaningful friendship. Another theme might be self - discovery. Maybe through their conversations and experiences together, both Beckett and Josh learn more about themselves.
Maybe they stumble upon a government conspiracy. The government wants to keep something hidden and they are seen as a threat, so they are made to disappear. It could also be related to a scientific experiment gone wrong. They are in the wrong place at the wrong time and get caught in an energy field that transports them to an unknown location, causing them to disappear from their normal world.
Another storyline might involve Beckett having a unique connection with his potential donor. Maybe the donor is someone he has met briefly before, and as he awaits the transplant, he has dreams or premonitions about this person. This could add an element of mystery and hope to the fanfic.
Well, it could be that Beckett is in a parallel universe where he has a different life and he's in bed with another man. In this story, they could be discussing their different worlds and how they got there. Maybe Beckett is an undercover agent and he has to pretend to be in a relationship with this man, so they are in bed just to keep up the pretense for some important mission. It could also be a story about Beckett's past life where he had a relationship with a man that he's now rediscovering while in bed with him.
Another possibility is that Beckett is trying to make someone else jealous. In the world of fanfic, there could be a complex web of relationships. He might be using this flirting as a tactic to get the attention of another character, perhaps someone he has a more complicated relationship with, like an ex - lover or a rival.