In such a fanfiction, the plot might involve Harry being an outcast even more than in the original series. Since he's the child of a Greek monster, the wizarding world might not understand him. He could struggle to fit in Hogwarts while also trying to figure out how to control his unique abilities that are a blend of wizardry and Greek monster traits. For example, if his 'parent' was a siren, he might have an alluring voice that he has to learn to use or suppress. It could lead to some really interesting adventures and challenges for him.
One possibility is that the story delves into the ancient Greek magic and how it interacts with the magic of the Harry Potter world. Harry could be on a quest to discover the origin of his Greek monster heritage and in the process, uncover secrets that change the way the wizarding world views magic. He might have to travel to ancient Greek ruins hidden in the wizarding world or face off against dark wizards who want to use his unique powers for evil.
The power dynamics in the story would shift. Harry's abilities as the child of a Greek monster could make him more of a target or a more powerful ally. Enemies like Voldemort might see him as an even greater threat. Also, his friends might view him differently, some might be in awe, while others might be a bit scared. And the whole concept of the prophecy could be re - interpreted in light of his new heritage.
It could be Athena. She is known for her wisdom, and Harry has shown great intelligence throughout the series. Maybe his strategic thinking and ability to solve complex problems come from his connection to Athena.
The plot might involve Harry being drawn into the world of Greek gods and goddesses. He could be on a quest to find out more about his mother, the Greek goddess. Along the way, he might meet other demigods or creatures from Greek mythology. His Hogwarts friends could be either helping him or being a bit confused by this new side of Harry. And he'd probably have to balance his life at Hogwarts with his new adventures in the Greek mythological realm.
He might also get involved in some sort of epic quest. The Greek Gods are known for setting heroes on grand adventures. Harry could be sent to retrieve a powerful magical artifact that has been lost for centuries, with the help or hindrance of the Gods. It could be a race against time as some dark force also wants the artifact.
Well, it could be that the Greek gods see Harry as a special being. They raise him on Mount Olympus. He would be taught by Athena in wisdom, so he'd be not only a powerful wizard but also extremely intelligent. His upbringing among the gods would give him a different perspective on good and evil. When he returns to the wizarding world, he'd use the values and powers he gained from the Greek gods to fight Voldemort. Maybe he'd use the gift of prophecy from Apollo to foresee Voldemort's moves.
It's really not a typical Greek Titan fanfiction. But if we force the connection, we could look at the family lines in Harry Potter. Some families have a long and powerful lineage, much like the Titans were part of an ancient and powerful family of deities. However, this is a very tenuous connection as the Potter family's story and the way magic is passed down is unique to the world J.K. Rowling created.
In such a fanfiction, Harry might discover new powers related to being a monster. For example, he could have enhanced strength or senses. Maybe he would struggle to fit in with his old friends at Hogwarts as they might be afraid of his new form.
In such a fanfiction, perhaps the plot could involve Harry seeing a child in a difficult situation, like an orphan in the wizarding world who has no family due to a dark wizard's attack. Harry, having been an orphan himself, feels a strong connection and decides to adopt the child. He then teaches the child about magic, love, and how to face the challenges in the wizarding world.
Perhaps the plot could be that the child of Harry Potter gets sent back in time accidentally. There, they meet a young Harry and his friends. The child has to be very careful not to disrupt the past too much while also trying to find a way back to their own time. This would be filled with lots of tricky situations, like when they almost reveal future events or when they have to avoid being seen by their own young parents.
It could be a young wizard who lost their parents during the final battle against Voldemort. Harry feels responsible for all the orphans created by the war and decides to take this particular child under his wing.