Well, their son could have a special ability that sets him apart. Maybe he can communicate with magical creatures in a way no one else can, which was a trait kind of hidden in both his parents. And the story could revolve around how he uses this ability to solve a mystery that has been plaguing the wizarding world. He might also struggle with the expectations that come with being the son of two such famous wizards.
The plot could involve their son being a key to uniting different wizarding factions. He could grow up in a world where there are still some divides left over from the previous battles. His very existence as the child of Harry and Hermione could symbolize hope for a more united future. He might travel around different wizarding communities, learning about their cultures and trying to bridge the gaps, all while facing some who oppose his mission due to jealousy or old grudges.
One possibility could be that their son inherits both Harry's courage and Hermione's intelligence. He might face new threats at Hogwarts, perhaps a new dark wizard rises. He could use his parents' knowledge and skills to outwit the enemy. For example, he might be great at Defense Against the Dark Arts like Harry and also be a top student in all academic subjects like Hermione.
It could be something like James, after Harry's father. It would be a way to honor his family's legacy.
One possible plot could be that Hermione and Harry discover a magical cat that has the ability to communicate with them. This cat leads them on an adventure to find a hidden magical artifact. Along the way, they use their knowledge and skills to overcome obstacles.
One possible plot could be that Harry and Hermione somehow end up in Westeros through a magical portal. They might use their magic skills from the wizarding world to deal with the political intrigue in Westeros. For example, they could help a certain house in the Game of Thrones - like scenario, using spells to protect their allies or uncover the secrets of their enemies.
One possibility could be that Harry and Hermione start a magical research project together. They explore some long - forgotten spells and their implications in the wizarding world. Maybe they discover a new way to use magic that could change the way magic is taught at Hogwarts.
One possible plot could be that Brandon has been keeping Harry's existence a secret for some reason. Maybe he's a powerful figure in a magical or special world and having a son would put Harry in danger. As the story progresses, Harry discovers his true parentage and has to deal with the new - found knowledge while also facing the challenges of this strange world he's now a part of.
Perhaps the fanfic could explore how Harry grows up in Thanos' domain. He could be trained in combat and in the use of the Infinity Stones. But as he gets older, he starts to question Thanos' motives. For example, when Thanos decides to snap his fingers, Harry could try to interfere and save half of the universe from disappearing, leading to a big showdown between father and son.
One possibility could be a moment of comfort. Maybe they are in a dangerous situation and Hermione is scared, so she sits on Harry's lap for a sense of security. Harry then consoles her and they share a deep, platonic moment.
One possible plot could be that Harry, initially shocked and confused, discovers his true parentage. Voldemort, instead of seeing Harry as an enemy, tries to groom him to be his heir. Harry might struggle with his new identity, having grown up believing he was an orphan with no relation to the Dark Lord. He could be torn between the values he learned at Hogwarts and the dark allure of his father's power. Eventually, he might use his unique position to try and change Voldemort's ways, or he could completely reject his father and fight against him in a new way, using his knowledge of Voldemort's weaknesses from their family connection.
Maybe they live together as part of a witness protection - like situation in the wizarding world. Some powerful and dangerous wizard is after them, so they have to stay together in a hidden, magically protected house. They learn to live with each other's quirks, like Harry's impulsive heroics and Hermione's strict study schedules, and in the process, they strengthen their friendship even more.
In a 'Harry and Hermione run away fanfic', they could be running away from the pressure of their pre - destined roles. Harry is always seen as the 'Chosen One' and Hermione as the brilliant sidekick. Fed up with these expectations, they decide to run away and explore the wizarding world on their own terms. They might travel to different wizarding schools across the globe, learning new spells and making new friends outside of Hogwarts.