I think Vernon Dursley can be really interesting in fanfic. He's such a rigid and angry character in the original, but in fanfic, he might have a hidden side. Maybe he has a secret love for magic deep down. And of course, Harry is always interesting in these fanfics. How he copes with the Dursleys in different scenarios can be very engaging to read about.
Definitely Dudley. In fanfic, he can be developed in so many different ways. He could become a wizard himself, or he could befriend Harry in a more meaningful way than in the original books. Another interesting character is Petunia. There could be more exploration of her past and her relationship with Lily, which can make her character more complex.
Petunia Dursley is also quite popular. There are fanfictions that delve into her past, her relationship with Lily, and how that affects her treatment of Harry. Some stories might show her having a change of heart and trying to make amends with Harry in a more complex way than just the basic storyline in the original books.
In many of these fanfics, there could be a new character introduced, like an old wizard who lives in the neighborhood secretly and notices Harry's situation. This character might play a crucial role in helping Harry deal with the Dursleys and his growing magical powers. He could be a mentor figure for Harry, guiding him through the tough times.
Well, Sirius Black could be an interesting character to put in a 'Harry Potter Dursleys fanfic'. Since he was Harry's godfather, his presence could stir things up for the Dursleys. He might be furious at how they treated Harry and take some sort of action. Hermione and Ron could also be included. They could visit Harry at the Dursleys' and be shocked at the living conditions. Their reactions and how they interact with the Dursleys could make for an exciting plot. And of course, the Weasley family as a whole. Their warm and loving nature would contrast greatly with the cold Dursleys.
A great 'Harry Potter Dursleys fanfic' might explore the Dursleys' backstory more deeply. For example, it could show how Vernon's hatred for magic and strange things developed over time. Or it could focus on Petunia's envy of Lily's magical abilities and how that affected her relationship with Harry. Some fanfics also imagine a scenario where the Dursleys are magically transported to Hogwarts and have to adapt to the magical environment. They would be completely out of their element, and it would be interesting to see how they interact with the students and teachers.
In this fanfic, the Dursleys may be depicted as being in over their heads. Vernon Dursley, for example, might be shown as constantly angry and trying to assert his authority in a situation where he has none. Petunia Dursley could be portrayed as being torn between her dislike for magic and her sense of family obligation towards Harry. Dudley might be shown as being bullied or affected in the Muggle world because of Harry's presence in their family.
Luna Lovegood often becomes an extremely interesting character in these fanfics. Her unique view of the magical world can lead to some really cool midnight escapades. For instance, she might lead Harry on a midnight search for a rare star - related magical object that is only visible at midnight.
Obviously, the Potions Master himself is very interesting. His complex personality, deep knowledge of potions, and often misunderstood nature make him a central focus. Then there are the students who interact with him, like Hermione. Her intelligence and eagerness to learn potions create some great scenes. Also, any original characters created by the fanfic authors can be really interesting, as they can bring new dynamics to the story.
Harry and Hermione are the main draws. Harry, with his courage and his new 'bound' state with Hermione, might show different aspects of his character. Hermione, being the intelligent and resourceful one, will surely have an interesting role in how their 'bound' relationship unfolds. Also, characters like Dumbledore could be fascinating. He might have some knowledge about their 'bound' situation and his guidance or lack thereof could be an interesting plot point.
It might show their fear in a more extreme way. Instead of just being dismissive of magic, they could be absolutely terrified of it. For example, every time Harry does something even slightly magical, they could cower in a corner.
In some of these fanfics, one interesting plot could be Harry finding solace in an unexpected friendship with a local magical creature that the Dursleys were unaware of. This gives him a sense of purpose and connection outside of the neglect he faces at home.