Maybe they attend a pre - Hogwarts magic class together. Harry struggles a bit with some of the advanced spells being taught, and Daphne, though initially a bit aloof, decides to help him. This could start a relationship where they learn from each other, with Daphne learning about Harry's unique experiences outside the wizarding world and Harry getting a better understanding of the pure - blood society through Daphne.
There could be a scenario where they are both at a wizarding library before Hogwarts. Harry is looking for books on his parents, and Daphne is there for some family - history - related research. They accidentally bump into each other in the aisle of old tomes. After some initial awkwardness, they start chatting. Daphne might share some interesting facts about the wizarding families in the area, and Harry could tell her about his muggle upbringing.
One interesting scenario could be them exploring hidden parts of Hogwarts together. They might find a secret room filled with old magic artifacts that have been long forgotten. This could lead to an adventure where they have to figure out how to use or protect these items from falling into the wrong hands.
Well, they might become friends. Maybe they discover they have a shared interest in magic even before getting to Hogwarts. For example, they could both be fascinated by magical creatures and exchange stories about the ones they've read about.
One possible plot could be that they meet in Diagon Alley. Maybe Harry is lost while shopping for his school supplies and Daphne helps him find his way. This could start a friendship where they exchange stories about their expectations for Hogwarts. It might also be a chance for Daphne to see Harry as more than just the famous Boy - Who - Lived, and for Harry to make a friend outside of the Weasleys and Hermione.
In such a fanfiction, Harry could die from an accident. For example, if he discovers magic early and while experimenting without proper guidance, something goes horribly wrong. Maybe he stumbles upon a dangerous magical artifact near his home that has a curse and it kills him before he ever gets to Hogwarts. It would be a very different story from the original, as his death would change the whole dynamic of the wizarding world that we know from the books.
One popular one might be 'A Different Beginning' which explores their relationship as they meet in a small wizarding village near their homes before Hogwarts. It shows how their initial encounters gradually turn into a budding romance.
One possibility could be that they meet in a muggle setting, like a local park. Maybe Hermione is reading a book on a bench and Harry, who is always curious, approaches her to ask what she's reading. Another could be that they both end up at the same orphanage for a short time and form a bond over their shared sense of being different from the other kids.
A magical creature could also be an important character. Imagine a friendly house - elf that they meet. The house - elf could be in some sort of trouble, and Harry and Daphne decide to help it. This adventure with the house - elf could shape their relationship and also introduce them to different aspects of the magical world.
In some fanfictions, they could be studying together in the library but also stealing glances and little touches. It adds a cute and romantic aspect to their relationship while still being in the Hogwarts setting. For example, Ginny might help Harry with his Potions homework, and they end up sharing a sweet moment over a difficult potion recipe.
One scenario could be that Daphne is bullied by some Slytherins for not conforming to their pure - blood ideals. Harry, having been bullied himself, steps in. He tells her that their opinions don't matter and that she is a great person regardless. He might even offer to practice defensive magic with her so she can stand up for herself.
Perhaps there was a dark wizard who was on the loose in the area where Harry lived. This wizard was attacking muggles randomly. Harry, being the Boy Who Lived with an inherent magical power, was able to disrupt the wizard's spell and it backfired, killing the dark wizard. This event could be a huge shock for Harry and his aunt and uncle, who have been trying to suppress his magic all along. It would also make his entry into Hogwarts even more complex as he has already had a major encounter with the dark side of the wizarding world.