One idea could be that the demigods from Percy Jackson's world end up in Forks for some reason. Maybe they are on a quest that leads them there. And they cross paths with the vampires and werewolves from Twilight. For example, Percy could use his water powers to fend off some danger that the Twilight characters are facing, like a wildfire started by a vampire's power going out of control. This would create an interesting dynamic as the two different mythologies collide.
One idea could be that the demigods from Percy Jackson's world have to team up with the vampires from Twilight to fight a common enemy, like an ancient and powerful dark force. Maybe a primordial being in the Percy Jackson universe has awakened and its influence is spreading, causing chaos even in the Twilight world, so they must combine their powers.
A cool plot might involve a character from the Percy Jackson series getting lost in the Kane world. For example, Percy could end up in the Kane world and not understand how their magic works. The Kane siblings would have to help him while also dealing with their own problems, like the threat of some dark magician. At the same time, Percy's friends from his world are trying to find a way to bring him back, leading to an interesting crossover of events and character interactions.
One possible plot could be that the Volturi from Twilight discover the existence of demigods from the Percy Jackson world. They see the demigods as a threat to their vampire supremacy. Percy and his friends have to team up with some of the more friendly vampires, like the Cullens, to fend off the Volturi. This would bring together the magic and powers of both universes in an epic battle.
One plot idea could be that Percy Jackson discovers a magical artifact that somehow links his world to the Marvel Universe. Thor senses the power of this artifact and comes to investigate, leading to a meeting with Percy. Meanwhile, the Avengers also get wind of it and think it could be a threat. So, they all end up in a big showdown where they have to figure out if they're on the same side or not.
One idea could be that the wizarding world discovers the existence of demigods. Harry and his friends team up with Percy and his crew to fight a common enemy that threatens both their worlds, like an ancient, powerful dark force that feeds on magic and the power of the gods. For example, a new kind of dark wizard who has made a pact with a minor, evil god.
Maybe there could be a new threat that can only be defeated by combining the magic of the Greek demigods and the Egyptian magicians. For example, a powerful monster that is immune to the weapons of one group but vulnerable to the powers of the other. Percy and his friends would have to team up with Carter, Sadie and their crew to take it down. They would need to learn about each other's worlds and powers along the way, which could lead to some interesting character interactions.
Another possibility is that Percy somehow gets transported to Eragon's world. He has to adapt to the new environment and figure out how his demigod powers work there. Eragon could initially be suspicious of Percy but then they realize they can help each other. Percy could teach Eragon about water magic, which is new in Eragon's world, and Eragon could show Percy how to ride a dragon, which would be an amazing experience for Percy.
A crossover could also involve a prophecy that links the two worlds. For example, a prophecy states that only the combined powers of a wizard from Hogwarts and a demigod from Camp Half - Blood can defeat an ancient evil that is rising again. Harry and Percy are the ones chosen, and they have to learn about each other's strengths and weaknesses in order to fulfill the prophecy. They could travel between the two worlds, recruiting allies from both sides.
Annabeth from Percy Jackson could be the most interesting. She's highly intelligent and strategic. In a crossover with Twilight, she would be the one trying to figure out how the vampires work and how to use their abilities to their advantage. She might also be a bit suspicious of them at first and would be constantly analyzing their every move, which could lead to some really engaging scenes in the fanfic.
Grover from Percy Jackson would also be a fascinating crossover. His connection to nature and his ability to sense things that others can't would be really cool in the Twilight world. He could befriend the werewolves because of their connection to nature as well. And his generally nervous but loyal personality would add a different flavor to the interactions in the Twilight world.