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.
Percy himself would be really interesting. His carefree yet heroic nature in the world of Twilight would be a contrast. He'd be the one bringing his demigod powers and his sense of humor. Just think of him trying to make friends with the vampires and werewolves while still being true to his demigod self.
Another great pairing could be Jacob and Grover. Jacob is a werewolf with a strong sense of loyalty and Grover is a satyr, always looking out for demigods. They both have a connection to nature in different ways. Jacob with his wolf form and Grover with his love for the wild. They could have some really fun adventures together as they explore both the Twilight and Percy Jackson worlds.
Percy Jackson and Thor would be an interesting pairing. Percy has his water powers and connection to the sea, while Thor has his lightning powers. They could have some really cool battle scenes together, like when they face a water - and lightning - based enemy. Also, they both have a bit of a sense of humor and a heroic nature, so they'd probably get along well.
Daenerys Targaryen would also be a top candidate. Her quest for the Iron Throne and her connection with her dragons. When she meets the demigods, she might see them as either threats or potential allies. Her experience with magic and her strong will would make for some intense interactions with the Percy Jackson characters, especially when it comes to issues of power and leadership.
I think Percy Jackson and Thor would make an interesting pairing. Percy has his power over water and is a skilled swordsman, while Thor has his mighty hammer and control over lightning. They both come from different worlds of power but have a lot in common. They could have some epic battles together against common enemies, like frost giants or sea monsters. And they could also exchange stories about their respective homes and families.
Nico di Angelo would be an interesting choice. His dark and brooding nature along with his powers over the dead would be a contrast to the more light - hearted aspects of the Harry Potter world in some ways. He could introduce the wizards to the concept of the Underworld and the spirits. And his ability to shadow - travel could be used in some really creative ways, like getting the group in and out of dangerous situations quickly, whether it's in Hogwarts or in the magical areas related to Greek mythology.
Definitely Percy and Carter. Percy is used to his own brand of heroism and power from the Greek world, while Carter has been trained in the ways of Egyptian magic. They could start off being a bit competitive, each thinking their own way is better. But as they face challenges together, they would learn to respect each other's skills. For example, when they are fighting a combined Greek - Egyptian monster, Percy's quick thinking in battle and Carter's knowledge of ancient spells could complement each other.
Annabeth and Sadie would be really interesting. Annabeth is super intelligent and strategic in the Greek way, while Sadie is a powerful magician with her own brand of cleverness from the Egyptian side. They could either be rivals at first, competing over who has the better knowledge or skills, or they could quickly realize that they can achieve more by working together. They could exchange ideas about architecture (since Annabeth loves it) and magic spells, creating a unique dynamic.
Hermione Granger and Annabeth Chase would make a great pair. Hermione's vast knowledge of magic and her intelligence could be combined with Annabeth's strategic thinking and knowledge of Greek mythology. They could work together to solve the complex problems that would arise in a crossover situation, like deciphering ancient spells or understanding the motives of the gods.