Another plotline could be that they are both in danger because of their family connection. Some evil force wants to use their relationship to gain power or access to something important. So they have to go on the run together, learning to rely on each other and their unique skills along the way. Maybe Percy has some magical abilities that Jacob doesn't, and vice versa, and they have to use these skills to outwit their pursuers.
A possible plotline could be that they inherit a mysterious family heirloom. This heirloom has some sort of power or significance in both the wizarding and Muggle worlds. Harry and Jacob have to team up to figure out what it is and how to use or protect it. Another plotline could be that Jacob gets caught up in a wizarding conflict by accident just because of his relation to Harry. Harry then has to protect him while also teaching him enough about the wizarding world to keep him safe.
There are various platforms where you can find such fanfiction. Archive of Our Own (AO3) is a great place to start. Just search for 'Percy is Jacob's cousin' in the search bar, and it'll show you relevant works. Some might be adventure - themed, focusing on their shared family bond while going on quests together.
One possible plotline could be that Percy has been betrayed by his closest friend. Maybe this friend revealed a big secret of Percy's, and now Percy feels isolated and heartbroken. He might start to question his relationships and his place in the world.
One possible plotline could be Percy's friends trying to break him out. They might have to overcome various obstacles like guards, traps, and security systems within the prison.
In this fanfiction, a plotline could involve Percy having to introduce Annabeth to his side of the family. This would be really interesting as it would show a whole new aspect of Percy's world to Annabeth. Another plotline could be Annabeth getting into trouble at school or with other demigods, and Percy having to step in as her father to resolve the situation. This would show how he uses his influence and authority in this new role. There could also be a plotline where Percy and Annabeth find out that there's a prophecy related to their new relationship. They would then have to figure out what it means and how to fulfill or avoid it, which would add a lot of tension and mystery to the story.
One possible plotline could be Percy's journey of grief and how he copes with the loss. He might become more withdrawn at first, then gradually find strength in his friends and his quest. Another could be his search for a way to bring his mom back, perhaps delving into dangerous magic or making deals with powerful beings. Maybe he also has to face the fact that his mom's death was part of a larger plan by the villains and he has to stop them from achieving their ultimate goal.
In a 'Jacob cheats on Bella with Leah' fanfiction, the plot might start with Jacob feeling neglected by Bella who is still somewhat hung up on Edward. Leah, on the other hand, is there for Jacob, and they gradually get closer. This closeness turns into an affair, and when Bella discovers it, there could be a big fight. Bella might be heartbroken and seek revenge, or she could try to win Jacob back. Meanwhile, Jacob has to deal with his guilt and the consequences within the pack as his actions could affect pack dynamics.
Perhaps Jacob could start to spread negative rumors about Bella among their mutual friends. This would create a rift not only between them but also within the larger group that they are a part of. He may feel betrayed by Bella for some reason and wants others to see his side.
Well, perhaps in the fanfiction, Annabeth betrays Percy's trust in a relationship way. She could start seeing someone else and use Percy's secrets against him to make herself look better in front of the new person. Or she might side with a group that Percy is against, believing their lies and false promises over Percy's warnings. This could lead to Percy feeling completely alone and having to find new allies.
Another plotline could involve Percy making new friends in the navy. These friendships could be tested during dangerous missions or when faced with internal conflicts within the naval unit.