He might learn about hidden magical talents in his family. For example, perhaps there were ancestors who could communicate with magical creatures in a unique way that was lost over time.
One interesting aspect could be how Harry's view of himself changes as he learns more about his family history. He might find out that his family had a role in some major magical events in the past, which gives him a greater sense of purpose and connection to the wizarding world. For example, if his ancestors were involved in the founding of Hogwarts or some great battles against dark magic, it would add a new layer to his identity.
His perception might become more positive. If he discovers great deeds his family has done in the past, he could feel a new sense of pride.
It might be a story where Harry delves into the family magic in order to solve some sort of mystery. For example, he could be looking for a way to defeat a new dark force and realizes that the family magic holds the key. He could learn new forms of magic related to protection or divination that are unique to his family.
One interesting thing could be how Harry discovers and masters Parselmagic in unique ways different from the original story. It might show his internal struggle with using a form of magic that has such a dark reputation.
In this type of fanfiction, a great plotline could be that Harry stumbles upon a hidden family grimoire. Inside, he finds magic that was thought to be lost. For example, there could be spells for creating protective shields that are stronger than any known spell. Or perhaps he discovers family magic that is related to time - travel, which he then has to learn to control carefully so as not to disrupt the timeline. This discovery of family magic would add a whole new layer to his character development and his role in the wizarding world.
It's possible that his ancestors were related to some of the more mysterious families in the wizarding world, like the Lovegoods. Maybe they shared certain magical abilities or eccentricities. They could also be linked to the Blacks, which would mean he has a connection to that complex family tree full of dark and light wizards.
Obviously, Harry Potter himself is a very important character as the story revolves around him learning family magic. His friends like Hermione and Ron might also be important as they could support or help him in his journey of learning this magic.
He could discover that he has a long line of powerful wizards. Maybe some of his ancestors were great duelists or inventors in the wizarding world. For example, they might have created spells that are now lost or forgotten.
It could be an older relative who was thought to be dead but actually has been living in seclusion, waiting for the right time to teach Harry about their family magic. This relative might have a vast knowledge of the family's magical traditions and spells.
In some fanfictions, Harry might learn family magic that has to do with time manipulation. His family could have had a special talent for seeing into the past or future. As he delves into this magic, he has to learn to control it carefully, as misusing time - related magic can have serious consequences. He may use this new - found magic to prevent certain disasters or to gain knowledge from the past that could change the course of the wizarding world.