Maybe a new, evil demigod who is jealous of Percy's status as the last of the seven. This demigod has made a deal with some dark forces and has powers that are a match for Percy's. He or she tries to undermine Percy at every turn and stop him from fulfilling whatever mission is associated with being the last of the seven.
It could be a corrupted demigod who has turned against his own kind. Maybe he was jealous of Percy's status as the hero and decided to use dark magic to become powerful enough to challenge him.
It could be a demigod who was once a hero but became corrupted by power. Maybe they were overlooked for some big honor at Camp Half - Blood and now they seek revenge on Percy and the other heroes. This character could have powers similar to Percy's but use them for evil, like causing tsunamis to destroy coastal cities instead of protecting them.
It could be a powerful demigod who has been corrupted by power. This demigod wants to be the sole champion of the arena and will stop at nothing to defeat Percy.
A powerful demigod who has been corrupted by the idea of primordial power could be the main antagonist. They might believe that by allying with a primordial or obtaining their power, they can become greater than the gods. This demigod would then oppose Percy, who stands for the protection of the current order.
A powerful Amazonian sorceress who has turned against her own kind and is now trying to use dark magic to take over the world. She could see Percy as an obstacle in her path and constantly try to defeat him.
Maybe a rogue demigod who blames Percy for something and wants to make his exile permanent. This demigod could have some dark powers or be in league with some evil forces. They would constantly try to thwart Percy's attempts to return or find a way to clear his name.
Maybe an ancient sea demon that has possessed a pirate captain. This possessed captain is now leading his crew to cause chaos everywhere. They are attacking other pirate ships and even some demigod - related places. Percy has to find a way to exorcise the demon from the captain and stop the crew's rampage. The demon is very cunning and uses the pirate's knowledge of the sea to its advantage.
The main antagonist could be Nyx. She is a powerful primordial goddess of the night, and Tartarus is her domain in a way. She might see Percy as an intruder and try to stop him from escaping or achieving his goals there.
It could also be a powerful sorceress who has her own motives for disrupting the demigod world. She might be seeking to gain control over the Olympian gods by using Percy as a pawn. She could use her magic to manipulate events and people around Percy, causing chaos and confusion.
Perhaps an ancient and dark deity who has been awakened due to some event related to Percy's exile. This deity wants to bring chaos to the world of gods and demigods, and Percy, even in exile, is the only one who can stop it. The deity would use all kinds of dark magic and minions to oppose Percy and his friends.