Well, in a lot of Petunia and Harry fanfictions, an alteration can be how their pasts are connected. Some fanfics might explore if Petunia had some sort of latent magical ability that she hid, and this could change how she views Harry's magic. It could also lead to her teaching Harry things about magic that she learned in her own way. Regarding Harry, his view of the Muggle world through Petunia could be different. He might come to understand and appreciate the Muggle side of his family more deeply, which is a big alteration from the original story where he was often at odds with the Dursleys.
Vernon is likely a character within the Psychonauts universe. In the context of the Vernon Ghost Story, he's probably the central figure who has some sort of encounter or relationship with the ghosts.
Vernon could be anyone. Maybe he's an old man in the neighborhood who loves to share spooky tales from his youth. Or perhaps he's a young person who had a strange encounter and turned it into a ghost story. Without more context, it's hard to say exactly who he is.
They can make the plot more complex. If Petunia is suddenly nicer to Harry, it changes the dynamic at the Dursleys' house. Harry might not be as eager to leave for Hogwarts, or he could have a different attitude towards Muggles in general. For example, if Petunia helps him with his magic studies secretly, he could be more advanced when he gets to Hogwarts, which would change his interactions with classmates and teachers.
Petunia could be a great addition. Her relationship with both Harry and Vernon adds a lot of complexity. She could be the mediator or the one causing more trouble between them, depending on the plot.
In some fanfiction, Uncle Vernon develops by becoming more accepting. For instance, he may start to see Harry not just as a freak but as a person with unique abilities that can be useful. He might begin to tolerate the wizarding world a bit more, especially if he realizes that Harry can use his magic to protect the family from certain threats.
Maybe Vernon is jealous of Harry's magical abilities. He might see Harry as a threat to his 'normal' family image. Also, Vernon has always been a bully at heart and Harry, being an easy target in his house, becomes the victim of his pent - up anger.
Maybe Dumbledore steps in at the last moment and takes Harry away from Vernon's house. He could use his powerful magic to protect Harry and remove him from that hostile environment.