A new, mysterious deity could be the main antagonist. This deity could be from a different pantheon and feels threatened by the power of Zeus and his son Percy. So, they start causing chaos in the world of the Greek gods and Percy has to stop them.
Another option is a group of rogue demigods who are jealous of Percy's status as a son of Zeus. They form a secret alliance and plot to bring him down. They could be led by a demigod with a grudge against the gods in general and sees Percy as a symbol of their power.
It could be a powerful Titan who escaped from Tartarus and wants revenge on Zeus by targeting his son Percy. This Titan might have unique powers that make him a formidable foe.
A powerful Titan who sees Percy as a threat to his plans. Since Percy has the power of Zeus and Poseidon, he could potentially stop any chaos the Titan wants to create. The Titan might try to turn Percy's friends against him or find a way to strip him of his powers.
A great antagonist could be a Titan who has a grudge against Zeus and sees Percy as an easy target to get back at him. For example, Atlas. He could try to use Percy to free himself from his burden again or trick Percy into doing something that would cause chaos among the gods. Another option is a powerful and vengeful demigod who feels that Percy, as the son of Zeus, has been given too much power and attention, so he wants to take Percy down.
A powerful Titan who was once imprisoned by Zeus and now wants to destroy his legacy. This Titan could be seeking revenge for his long imprisonment and sees the legacy as a symbol of Zeus's power that he wants to crush.
One idea could be Percy having to mediate between Zeus and Hades during a major feud. Since he's the son of both in this fanfiction, he has a unique connection to them. He could use his powers and his knowledge of both of their domains to try and broker peace.
Hera could be a main antagonist. Since she's often jealous and protective of Zeus, she might not be happy with Percy being adopted. She could try to undermine him at every turn. Another could be Ares. He might see Percy as a threat, especially if Percy starts to gain more power as Zeus' adopted son.
Some of the minor gods who feel overshadowed by Zeus and Hera might be antagonists. In this fanfiction, they could see Percy as a symbol of the over - dominance of Zeus and Hera. They might plot against Percy to send a message to his parents. These minor gods could use their unique powers and their knowledge of the magical world to create all sorts of problems for Percy, from cursing him to setting up traps in his path.
Yes, there are quite a few interesting ones. You can start by looking on fanfiction platforms like Archive of Our Own. Some fanfics might explore Percy's relationship with his divine parents in unique ways, or perhaps put him in new and exciting adventures that are different from the original series.
Perhaps an ancient and evil creature that has a grudge against Oin. Since Percy is Oin's son, this creature targets him. It could be a monster that was once defeated by Oin and now wants revenge by going after his son.
It could be the Titans who see Percy as a threat. Since he is the son of Kronos, they might want to either use him for their own purposes or eliminate him to prevent any interference in their plans.
Another option could be a corrupted demigod. Maybe a child of a minor Egyptian god who is jealous of Percy's status as Son of Sobek. This demigod could form an alliance with some dark forces and try to undermine Percy at every turn, creating all sorts of challenges for him.