Perhaps the original family had a hidden treasure or a powerful magical artifact. In the fanfiction, Harry and his friends could go on a quest to find it. Along the way, they discover that the original family had ties to other magical creatures, like centaurs or merpeople, and these relationships are crucial in unlocking the location and power of the artifact.
One idea could be that Harry is placed with a family of magical creatures protectors. They teach him about the hidden magic in nature and the importance of protecting all magical beings. This could lead to Harry having a different view on magic compared to what he learned at Hogwarts.
One idea could be that Harry and Hermione get married and have a family in a world where Voldemort was defeated much earlier. Their children could inherit both their magical abilities and intelligence, and they might face new threats to the wizarding world that require their combined skills to overcome.
Another plot could focus on how the family magic affects relationships. Maybe Harry uses family magic to strengthen his bond with his children in a way that normal magic can't. He could teach them unique family spells that help them understand their heritage better.
A more complex plot could be that the entire Family Guy cast gets transported to Hogwarts during a school year. Quagmire could be the new Defense Against the Dark Arts teacher, bringing his own unique brand of 'defense' which mostly involves flirting with the female students. Lois could get into trouble for trying to organize a Muggle - style PTA at Hogwarts, and Peter could end up on the Quidditch team as an accidental addition.
One idea could be that the Potter family totem is a phoenix. It represents rebirth and magic, which are very much associated with the Potter family. Harry himself has a connection to the phoenix Fawkes.
Another plot idea could be a family that adopts a magical creature as a part of their family. In the wizarding world, this is quite unusual. The family then has to deal with the reactions of the wizarding community. They also have to learn how to take care of the creature, and the creature could end up helping the family in unexpected ways during their adventures in the Harry Potter - like world.
One idea could be exploring Harry's family vacations. Maybe they go to a magical beach where the kids learn special water - based magic. Another could be a story about Harry teaching his kids about the Marauder's Map and the adventures they have trying to find hidden passages in Hogwarts that even he didn't know about.
Well, perhaps Harry is adopted by a family of magical creatures, like centaurs. They teach him their unique ways of magic and understanding of the magical world. He could have a whole different perspective on things like divination, as centaurs are known for their prophetic abilities. It would be really interesting to see how he interacts with his human friends at Hogwarts after growing up in such a different environment.
Another plot could involve a member of the royal family in the Muggle world being a Squib. This person, desperate to be part of the magical world, tries to find a way to regain their magic through the help of the Potter family. The story could then revolve around the secrecy that needs to be maintained while also trying to help this person, and how it affects the relationship between the wizarding and Muggle royal families.
You could also focus on the Weasley - Potter family relations in a fanfic. For instance, how the two families' magic combined in different ways. Maybe there are some spells that only work when a Potter and a Weasley work together, and exploring those spells and the situations where they are used can make for a great fanfic story.