Some fanfictions with core threads deal with the concept of magic itself as a core thread. They might explore how magic could be used in more creative ways or what would happen if certain spells had different effects. For instance, a story could be about a spell that backfires and changes the very nature of Hogwarts. Also, there are fanfictions that take the core thread of the characters' family backgrounds. For example, exploring more about Harry's parents' lives before they died or what Ron's family was really like in their early years.
One great one could be those that explore the core thread of friendship. For example, stories where Harry, Ron, and Hermione's friendship is put to the test in different ways than in the original books. Maybe they face a new dark magic threat that makes them question their loyalty to each other. Another type might be fanfictions that focus on the core thread of Hogwarts itself. They could describe new secrets within the castle walls, or different ways the houses interact. There are also those that take the core thread of the wizarding world's politics and expand on it, showing how different families and factions vie for power in ways not fully explored in the main series.
Well, a popular core thread in Harry Potter fanfiction is the prophecy. There are many fanfictions that revolve around the prophecy about Harry and Voldemort. They might change the interpretation of the prophecy or show what would have happened if the prophecy was never made. Some fanfictions also focus on the core thread of the Marauders. These stories often go back in time to show the adventures of James, Sirius, Remus, and Peter during their Hogwarts days, which gives more depth to the characters that we only know from the memories in the original series.