They could be related through a shared ancestor in a really convoluted backstory. Imagine there was an ancient being who had descendants in both the world of Greek mythology (where Percy Jackson is from) and the wizarding world. This being's bloodline could have led to Percy on one side and Dumbledore on the other, perhaps giving them both their unique powers and a connection that's yet to be fully discovered.
One possible plot could be Percy discovering his relation to Dumbledore. Maybe he starts having strange dreams or visions that lead him to the wizarding world. He then has to learn about his new family connection while also trying to keep his demigod identity a secret from most of the wizards, as they might not understand it.
In fanfiction, they could be related via a magical curse. A curse placed on Percy's family in the past by a dark wizard related to Voldemort could be the link. This curse could have some strange effects that tie their fates together. For example, Percy might be the only one who can break the curse and defeat Voldemort. Another way is if there is a secret magical society that both Percy and Voldemort are unknowingly part of, and the rules of this society create a relation between them, like they are both heirs to some powerful magic within the society.
This could be a really interesting fanfiction idea. Percy Jackson has his own rich world of Greek mythology in the 'Percy Jackson' series, and Dumbledore is from the 'Harry Potter' universe. Combining them could lead to some unique magic and adventure. Maybe Percy would inherit some of Dumbledore's wisdom and magical abilities along with his own demigod powers. It could be set in a world where the two fictional universes collide, and Percy has to use both his skills to face new threats.
In fanfiction, they might be related through a shared enemy. Let's say there's a new super - villain who is causing chaos in both the demigod world and Gotham. Percy and Bruce start to work together to stop this foe, and during their investigations, they find out that there's a family connection between them. It could be something as simple as an ancestor who was part of a secret society that had ties to both the Greek mythology realm and the early settlers of Gotham.
Percy would be really angry. He's used to being in control of his own situation, and being kidnapped is not something he would take lightly. He'd probably try to fight back right away, using his demigod powers like his sword fighting skills and his ability to control water.
A possible relationship could be that of mentor - mentee. Dumbledore could see Percy's potential and take him under his wing, much like he did with Harry Potter. This would be an interesting dynamic as Percy has his own set of skills and powers from the Greek mythology world.
Well, in a fanfiction scenario, if Dumbledore kidnapped Percy Jackson, it could lead to a really strange and interesting mix of the two universes. Dumbledore might be interested in Percy's powers, especially his connection to the water and his demigod abilities. Maybe he wants to protect Percy from the threats in Percy's world by taking him to Hogwarts. But Percy would surely be confused and probably try to escape to get back to his friends and his own world.
Percy could start to form new relationships within the Faunus community. These relationships could teach him different values and ways of life. Maybe he'd meet a Faunus mentor who would help him understand and control his new Faunus powers, if he has them. This would be a big part of his development, as he'd be learning from a completely different perspective than he's used to at Camp Half - Blood.
No. Percy Jackson is a character from Rick Riordan's series which is based on Greek mythology, having no relation to Dracula fanfiction.
Percy could be developed by facing moral dilemmas. For example, when dealing with the eldest Titan, he has to decide whether to trust this ancient and powerful being. This forces him to grow and think more deeply about right and wrong.