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.
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.
One possible fanfiction could be about Petunia Dursley having a secret love for lemon - flavored treats and how it changes her mood. For example, she discovers a new lemon cake recipe and it makes her momentarily forget her usual grumpiness.
Some fanfictions might involve Petunia Dursley's relationship with lemons in a more comical way. Maybe she tries to make a lemon - based household cleaner that ends up having some very unexpected and humorous results, like turning the whole kitchen a bright yellow color or making a strange smell that attracts magical creatures.
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.