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.
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.
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.
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.
In a more comical plot, the lemon could be a mischievous magical object that keeps causing chaos in Petunia's house. It might make things float or turn into strange creatures. Petunia, being the uptight person she is, has to deal with this lemon - induced mayhem while also trying to keep up appearances in front of the neighbors. This could lead to some really funny situations and a change in her attitude towards the strange and magical over time.
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.
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.
One common theme is the exploration of the divide between the Muggle and wizarding worlds. Since Petunia is a Muggle - born adjacent character and Sirius is a pure - blood wizard, their interactions often highlight the differences and similarities. Another theme is redemption. Sirius may be seen as trying to make up for past mistakes through his relationship with Petunia.
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.
A possible plot could be a time - travel scenario where Harry and Petunia are sent back in time. They have to rely on each other to survive in the past, and in the process, they learn more about their family history and each other. This shared adventure could lead to a deeper understanding and a different kind of relationship between them.