Well, he might get involved in the supernatural chaos of Beacon Hills. Maybe he'd team up with the local werewolves or other supernatural beings to fight against a common enemy. He could use his demigod powers in unique ways, like using his water control against a fire - based threat in Beacon Hills.
There's also a chance that he'd meet Lydia first. Lydia has her banshee powers, and she might be drawn to Percy because of his otherworldly presence. She could sense something about him that's not of this world, and approach him out of curiosity. Their meeting could be a very intense and mysterious one, with Lydia trying to figure out what exactly Percy is.
The Nogitsune could also be a great antagonist. It thrives on chaos and mischief. Percy's presence and his powers would be something the Nogitsune would want to either corrupt or use for its own evil purposes. It could try to possess Percy or those around him to gain access to his demigod abilities.
One idea could be Percy arriving in Beacon Hills and being mistaken for a new supernatural threat. The local pack tries to figure out his powers and intentions. Another could be Percy teaming up with the pack to fight a common enemy that is drawn to the unique energy of Beacon Hills and Percy's own demigod powers. Maybe he also has to adjust to the high school life there while hiding his true identity from most of the students.
If Harry Potter goes to Beacon Hills, he'd probably be quite confused at first. The rules of the supernatural world there are different from the wizarding world. But then, he would start to adapt. Maybe he'd team up with Scott McCall. Harry's experience in fighting Voldemort would give him the courage and skills to face the threats in Beacon Hills, like the Dread Doctors or the Nogitsune.
Well, Hermione is a strong - willed character. In Beacon Hills, this would serve her well. She would not be afraid to stand up to the powerful supernatural forces there. She could become a sort of mentor figure to the younger characters. She might teach them the value of knowledge and how to use it in the face of danger. Also, her magic could be a great asset in battles against the various threats that the town faces on a regular basis.
Percy going to Ilvermorny would be a wild ride. First, he'd be amazed by the castle itself. The four houses would be a new concept for him. He might be sorted into a house based on his heroic nature, like Thunderbird perhaps. He'd introduce the idea of Greek gods to the students there, which would cause quite a stir. And his relationship with Annabeth would be put to the test as they are separated by different magical institutions.
He might meet some talented witches and wizards. Maybe there would be a student who is an expert in magical beasts and they could share stories about their encounters with strange creatures.
Percy might meet an old enemy who has somehow been resurrected or changed in the future. This enemy could be more powerful and have a new plan to take over the world or destroy the demigods. Or he could meet a group of time - traveling demigods from an even further future who are trying to fix something in Percy's future that has gone wrong.