In many 'Harry Potter Dursley fanfiction' stories, the Dursleys are used as a contrast to the wizarding world. Their normal, mundane lives are juxtaposed with the magic and wonder of Harry's world. This can create interesting storylines where they are either forced to accept the magic or try to fight against it in their own stubborn ways.
By focusing on their shared past traumas. For example, both have had experiences related to Lily's death in different ways. Harry lost his mother, and Petunia lost her sister. They could start to bond over this common pain and gradually move towards a more positive relationship.
There might be a fanfiction where the Dursleys are forced to enter the wizarding world due to some magical accident. This could lead to all sorts of comical and tense situations as they struggle to adapt to a world they have always despised. For example, Vernon Dursley having to deal with talking portraits or flying broomsticks.
Petunia Dursley is significant as she represents the non - magical world's reaction to magic in the Harry Potter series. She is a Muggle who has a complex relationship with the magical world. She has seen magic through her sister but resents it. Her actions towards Harry, like treating him as an outcast in her home, show how some Muggles might fear or be envious of the magical abilities they don't possess.
The popularity also lies in the fact that it allows fans to expand on the 'Harry Potter' universe. By focusing on the relationship between Harry and Dudley, fanfiction writers can create unique stories that might include elements that were only hinted at in the original series. For example, exploring how Dudley's upbringing in a Muggle family affected his view of Harry's magic, and how that view could be altered in different scenarios.
One possible fanfic could be about Petunia Dursley finding a magical toy that once belonged to Harry Potter's parents. Maybe she discovers it in the attic and it starts to show her memories of Lily and James Potter. She realizes that she has been wrong about Harry all along and decides to change her attitude towards him.
One interesting 'Harry Potter Dursley fanfiction' could be about Dudley Dursley having a change of heart after the war. He realizes how cruel he and his family were to Harry and tries to make amends. Maybe he starts to study magic in secret, trying to understand the world he once shunned.
One popular 'Harry Potter Dursley' fanfiction might be 'The Dursleys' New Perspective'. It delves into how the Dursleys' lives are changed in unexpected ways. Another could be 'Dursley Encounters', which explores their interactions with the wizarding world in a unique manner. There's also 'The Secret of the Dursleys' that reveals hidden aspects of their family.
I'm not sure if there are 'popular' ones in the sense that they are widely known and accepted. Lemon fanfictions are a bit of a niche and often not mainstream due to their mature content. But there might be some that have a following within the more hardcore Harry Potter fanfiction community.
There might be some out there, but it's a rather unusual pairing considering Vernon's often cruel treatment of Harry in the original series. Fanfiction can take all sorts of creative turns, though. You could try searching on popular fanfiction websites like Archive of Our Own or Fanfiction.net. However, this pairing goes against the typical relationship dynamic presented in the Potter books.