A group of power - hungry elders within the demigod community could be the antagonists. They fear the growing influence of Percy and Annabeth among the younger demigods. They banish the two in order to maintain their own power and control. These elders might have their own hidden agendas and are willing to sacrifice Percy and Annabeth to achieve them.
The main antagonist might be an ancient and powerful being that has awakened. This being could see Percy and Annabeth as a potential threat to its plans of taking over the demigod world. So, it uses its influence to have them banished, thinking that without them, it would be easier to conquer. For example, a long - forgotten Titan who has found a way to break free from its prison.
The main antagonist might be a new character, a powerful being that rules the realm where Percy is banished. This being could see Percy as a threat to his or her power and try to stop Percy from finding a way back.
A powerful sorcerer who also wants to claim the island for his own could be the main antagonist. He sees Percy as an obstacle to his plans and tries to eliminate him. The sorcerer might have control over some of the dangerous creatures on the island and use them to attack Percy.
A group of rogue demigods could be the main antagonists. They might be jealous of the attention the daughter gets because of her parents. So, they plot to make her life difficult and try to drive a wedge between her and the other demigods.
It could be a rogue demigod who has a vendetta against Percy. This demigod might have spread the false rumors that led to Percy's banishment in the first place, and now he wants to make sure Percy never returns. He uses his own powers and manipulates other demigods and monsters to achieve his goal.
It could be an old enemy of Percy and Annabeth. Maybe a monster that they thought they had defeated long ago but has come back for revenge. This time, it might target Annabeth's daughter to get back at them.
Maybe an old and powerful monster that has long been trapped in Tartarus could be the main antagonist. It sees Percy as a threat or a source of entertainment, and it tries to stop him from finding a way out or from getting stronger.
Another possibility is a group of outlaws or rebels in the banished world who don't trust Percy because he's from another place. They might think he's been sent to spy on them or take over. So they would constantly try to get rid of him. They could be very resourceful and know the ins and outs of the new world better than Percy, which would make them tough opponents.
One possibility could be that they are banished due to a false accusation. Maybe someone in the camp frames them for a serious crime they didn't commit, like stealing a powerful magical artifact. So, they are forced to leave and then have to prove their innocence while on the run from those who banished them.
Maybe the main antagonist is an old enemy of Percy's who has learned about his newfound ability to hypnotize (even if it was accidental). This enemy decides to use Annabeth as a pawn by hypnotizing her, hoping to lure Percy into a trap and defeat him once and for all.
A corrupted demigod who has turned to the dark side could also be the main antagonist. This demigod has made a deal with the forces of the House of Hades in exchange for power. They envy Percy and Annabeth's relationship and their status among the demigods, so they try to destroy them. They know all the secrets of the House of Hades and use this knowledge to constantly stay one step ahead of Percy and Annabeth.