In some fanfictions, Petunia's character develops from being a cold and resentful aunt to a more understanding and protective figure. For instance, when she learns more about the wizarding world's threats to Harry, she starts to see him as more than just an unwanted burden.
Often, Petunia's development is linked to her relationship with her sister Lily. Fanfictions might show her reconciling with Lily's memory through her connection with Harry. She could start to embrace the magical part of her family history she had long rejected, which is a significant character shift.
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 are fanfictions where Petunia has a change of heart towards Harry. For example, she realizes the error of her ways after seeing how much danger Harry is constantly in at Hogwarts. She then tries to make amends by secretly helping him, like sending him care packages with useful Muggle items that can be adapted for magical use.
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 could be a fanfiction where Petunia Dursley discovers some hidden magic within herself and this somehow links her more closely to Harry. In this scenario, the story could take on a whole new dimension. But as for a particular 'Harry Potter and Petunia Dursley lemon fanfiction', it's hard to point out one without really searching through various fanfiction sites. Also, remember that lemon content often implies sexual undertones which may not be appropriate for everyone.
Maybe she secretly has a soft spot for the magical world she once shunned. Fanfiction might offer her a different perspective on Harry's life, one that she can explore without fully admitting her growing curiosity or even a latent affection for the boy she once treated so poorly.
She might like the ones that focus on Harry's early years in the Dursley household. Stories that show his struggle from her point of view could be appealing to her. It would be interesting for her to see how others imagine her thoughts and actions during that time.
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.
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.
You can start by looking on popular fanfic websites like Fanfiction.net or Archive of Our Own (AO3). Just type in 'Harry Potter Petunia Dursley' in the search bar, and a whole list of relevant fanfics should pop up. These sites have a large collection of fan - created stories, so you're likely to find something that suits your taste.
A fanfic could connect them by having a toy that was passed down from Lily Potter to Harry, but Petunia has kept it hidden all these years. When Harry discovers it, it causes a big conflict as Petunia doesn't want to let go of the last piece of her sister she has in that way. But eventually, it could lead to a moment of reconciliation between them.