Apollo could be Harry's father. Apollo is the god of music, poetry, and prophecy among other things. If Harry was Apollo's son, he could have an innate talent for divination that is far beyond what he showed in the original books. He might also be able to use music as a form of magic, perhaps charming his enemies or healing his friends with melodies.
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.
One possible plot could be that Harry is transported to the world of Greek gods. There, he finds out that the Greek gods have been observing the wizarding world for a long time. They believe Harry has a special role to play in saving both their worlds from a common enemy. So, Harry has to adapt to the new environment, learn about the powers of the Greek gods, and find a way to combine his magic with their divine powers to defeat the enemy. He might also make some friends among the Greek demigods and gods, and there could be a lot of interesting interactions between him and the well - known figures of Greek mythology like Zeus, Athena, etc.
The plot could involve Harry being torn between the two worlds. He has his loyalty to Hogwarts and his friends there, but his Greek god side pulls him towards a different destiny. He could be on a quest to find out which Greek god is his father. Along the way, he might meet other demigods who help him understand his powers better. And he could use his combined knowledge of magic and godly abilities to defeat a great evil that threatens both the wizarding and Greek mythological realms.
In some fanfictions, it could be Sirius Black. Some fans like to imagine that Sirius had a deeper connection to James Potter's family than just being best friends. They might create a scenario where there was some sort of magic or prophecy involved that made Sirius more than just Harry's godfather, but his actual father in this alternate story line.
Perhaps it could be Thor from Norse mythology. Thor is the god of thunder. So, Harry could have the power to control lightning or storms. He would be a very formidable wizard with such powers. His Hogwarts years would be full of situations where he uses these thunder - related powers to solve problems or fight off enemies.
Delia could also be a character created entirely for the fanfic. She could be an old friend of Dumbledore's who was entrusted with the task of protecting Harry. Dumbledore knew that she had a special connection or power that could keep Harry safe from Voldemort. So, in this case, Delia is a key figure in Harry's life from the very beginning, but her identity was kept secret until the right time.
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.
One interesting concept could be how Harry's new heritage as the son of a Greek god might change his magic. For example, he could have access to different types of magic that are more related to Greek mythology, like the ability to control the elements in a more direct way. Another concept might be his relationships with the other characters. Maybe he'd have a different dynamic with Hermione and Ron, as they try to understand this new side of him. Also, it could introduce new enemies or allies from the Greek mythological realm into the Harry Potter universe.
Maybe Ron Weasley. Imagine a fanfiction where Ron, after getting married, has triplets. Given his humorous and sometimes clumsy nature, it would be really fun to see how he and his wife deal with three little ones. He could pass on his love for food and his loyalty to his children in a really endearing way.
One possibility could be that Harry is the son of a powerful but mysterious wizard and a muggle - born witch who had to give him up for some reason. Maybe his real father was a dark wizard who was hunted down, and his mother was a brave witch who wanted to protect Harry from that dark legacy.