Percy might run into Loki. Loki is known for his trickery. He could either befriend Percy for his own devious plans or try to deceive him. Percy, being used to dealing with tricky situations in his own world, would have to be on his guard against Loki's schemes.
Well, he might meet some really tough Norse warriors. Maybe he'd get into some epic battles or challenges. He could also make new friends or enemies among the Norse gods and heroes there.
If Percy Jackson goes to Valhalla in fanfiction, he could meet a whole bunch of Norse heroes. There's Sigurd, the dragon - slayer. And then there are the Valkyries who choose who gets to go to Valhalla. He might also run into Loki, who is always causing some sort of trouble. Percy would be in for quite an adventure as he tries to figure out who he can trust among these new and often unpredictable characters.
Well, one possibility could be that Percy discovers a new aspect of the mythological world. Maybe in this fanfic, his death is a sacrifice to save his friends. Once in Valhalla, he could meet new heroes from different mythologies and form alliances or rivalries. He might also try to find a way back to life or to his friends in the world he left behind.
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.
In this fanfiction scenario, Percy's adaptation to Hogwarts would be a complex process. He'd first have to deal with the cultural shock of a new magical world. The Hogwarts houses would be different from the cabins at Camp Half - Blood. He'd need to figure out which house he belongs in. His skills with a sword might not be as useful as in his own world, but he could learn to use magic to enhance his combat abilities. Also, he'd have to make new friends and enemies, and learn how to navigate the social politics of Hogwarts while still holding on to his own identity as a demigod.
Some of the more traditional and rule - following demigod groups at Camp Half - Blood could be against him. They believe in following the established ways and Percy's rogue behavior goes against that. They might see him as a troublemaker and try to bring him back to the 'right' path.