Well, they could start off by being quite surprised to see each other. Maybe they meet in the library at Hogwarts. Harry might be looking for a book on Defense Against the Dark Arts and Daphne could be there studying potions. They could exchange a few words, like Harry asking Daphne if she's seen a certain book. And then they might realize they have more in common than they thought.
One possibility is that they meet at a Quidditch match. Daphne could be there to support her house team. Harry, being a great Quidditch player, catches her eye. After the match, they meet up. Daphne might compliment Harry on his flying skills, and Harry could invite her to have a butterbeer with him. They could then start a friendship that might develop into something more as they get to know each other better.
In the fanfic, perhaps they meet outside of Hogwarts during the holidays. Harry could be on a walk in a wizarding village when he bumps into Daphne. They could end up having a conversation about the wizarding world's future. Daphne might share her thoughts on how pure - blood and half - blood wizards can work together better, and Harry could be really interested in her perspective.
In the fanfic, it's possible that they are both there on a secret errand. Harry could be looking for a hidden magical item that he believes is stored in Gringotts for safe - keeping. Daphne might be there under orders from her family. When they meet, they might initially be guarded about their true purposes. However, as they interact, they start to trust each other and decide to collaborate, which could set off a whole new chain of events in the story.
Perhaps they exchange books right away. Harry shows Daphne his spell - books, and Daphne offers him one of her advanced potions books. This simple act of sharing could spark a connection between them. They might then sit together for the rest of the train ride, chatting about the upcoming school year and speculating on what new adventures and challenges await them at Hogwarts.
A common plotline could be that they meet in a secret place in Hogwarts, like a hidden room. They discover they both have secrets they can't share with others easily. For example, Daphne might be involved in some kind of research on a rare magical plant, and Harry could be having some strange dreams about Voldemort again. They bond over their need for secrecy.
In the 'Daphne meets the Dursleys fanfiction', it's possible that Daphne initially has a hard time understanding the Muggle - like ways of the Dursleys. However, as time passes, she might start to see through their facade. She could then form a plan to get Harry out of that unpleasant environment. For example, she could pretend to be a social worker or something similar in the Muggle world and convince the Dursleys to let Harry go with her. And then she could introduce Harry to the wizarding world in a more positive and gentle way than how the Dursleys have been treating him.
One possibility is that their meeting is not so friendly at first. Daphne, coming from a certain Slytherin - pure - blood mindset, might be a bit haughty towards Harry. But as they talk, Harry's kindness and honesty could break through her initial prejudice. They could end up having a deep conversation about the differences between the houses at Hogwarts, and how they both want to make the most of their time at school, despite their different backgrounds.
In this fanfic, perhaps Harry could have a really heart - warming conversation with the Grangers. He might learn a lot from them about the Muggle world, like how they approach education and family values.
Well, perhaps in the fanfic, there's a big event in the Great Hall. Harry and Daphne could get paired up for some sort of task or competition. This would force them to interact and get to know each other better. They might start to see each other in a new light, beyond the usual Hogwarts house divides. As they work together, they could discover that they have more in common than they thought, like their love for a particular subject or their shared sense of adventure.
They could become friends. Harry, being the hero he is, might reach out to Daphne, who is a Slytherin. At the Yule Ball, they could share a dance and discover they have a mutual interest in magical history. This could lead to them spending more time together, studying in the library and sharing their different perspectives on various magical topics.
Well, it could be really interesting. Maybe Q would use his omnipotent powers to help Harry in his fight against Voldemort. Q could transport Harry to different places instantly, or provide him with some super - powerful magic items that are beyond the normal wizarding world.
In a fanfic where Harry Potter meets Death, one possibility is that Death offers Harry a choice. Maybe it's a choice between saving his loved ones or sacrificing something of great value to him. This could lead to a moral dilemma for Harry, testing his character in ways not fully explored in the original series.