He'd probably have a different set of enemies. In Ancient Greece, the threats might be more centered around the local myths and power struggles. For example, he could be in the middle of a feud between two powerful city - states, each with their own godly patrons.
His birth could introduce new prophecies. For example, the oracles might start predicting things related to Percy's fate that could change the course of events in Greek mythology.
His life would change drastically. He would be a being of immense power. He would no longer have to fear the wrath of the Olympians as he would be on a similar or even higher level. He could shape the world as he wished.
Percy's relationships would be in upheaval if he were born a werewolf. His friendship with his demigod comrades at Camp Half - Blood would face many challenges. Some might be accepting, but others would fear his new - found abilities and the potential danger he poses. His relationship with his father, Poseidon, would also be altered. Poseidon might see Percy as more of an abomination or a unique tool depending on his own complex nature. Percy's love life, like with Annabeth, would be full of complications. Annabeth would have to decide if she can love a werewolf - demigod hybrid, and Percy would struggle with his own self - image in relation to her. His relationships with other mythological creatures, such as centaurs and nymphs, would also change as they would view him through the lens of his werewolf nature rather than just his demigod status.
Well, in a fanfiction where Percy is born in Greece, his powers would likely have a more 'Greek - flavoured' aspect. Since he's in the homeland of the Greek gods, his connection to water could be intertwined with the Greek concept of the four elements. So, his water powers might interact more with the other elements in a way that is more in line with Greek elemental theory. He might also be able to draw on the power of Greek water deities more directly, which could enhance his abilities and give him new ways to use his powers in battles and other situations.
If Percy Jackson was born in Greece, his story would be filled with even more direct connections to Greek mythology. He'd grow up surrounded by the ancient sites and stories. Maybe he'd be discovered earlier by the demigod community there. His powers could be more in tune with the original Greek magic, and he might have different relationships with the Greek gods who are closer to home, so to speak.
His life would change drastically. He'd be living in a different environment, like the Batcave perhaps. He'd have to adapt to Batman's strict rules and training regime.
Percy's daily life would be different. Instead of going to Camp Half - Blood on land, he might attend an Atlantean school for demigods or those with special abilities. He would also have to adapt to the Atlantean diet which could be very different from what he's used to. And his friends would likely be a mix of Atlanteans and other demigods who have also made their way to Atlantis.
If Percy becomes a werewolf in fanfiction, his social life would be turned upside down. His friends at Camp Half - Blood might be hesitant to be around him, fearing his new, more unpredictable nature. He might be drawn to other werewolves or creatures of the night, which could introduce him to a whole new set of allies and enemies. Also, his training as a demigod would need to be adjusted to account for his werewolf abilities, like learning to fight in his new form without losing control.
He could have adventures like battling mythical creatures. For example, fighting the Minotaur in the labyrinth. Maybe he'd also encounter the gods and goddesses and get involved in their power struggles.
First, do your research on Ancient Greek mythology. Know the gods, the monsters, and the heroes well. For example, if you're writing about Percy facing the Minotaur, you need to know all the details about it from the original myths. Second, stay true to the characters of the Percy Jackson series. Percy is brave but also has a sense of humor, so keep that in his character. Third, create an engaging plot. Maybe it could be about Percy finding a lost temple in Ancient Greece that holds a powerful secret.