Well, it could be that the Daily Prophet has printed something about Harry's family that is untrue and deeply offensive. For example, they might have dug up some false information about his parents and their relationship. Harry, being protective of his family's legacy, decides to take legal action against the newspaper. This could lead to a whole series of events in the fanfic, such as him having to gather evidence, find a wizard lawyer (if there is such a thing), and face the powerful influence of the Daily Prophet's owners and editors in the wizarding world.
Maybe Harry sues the Daily Prophet for spreading false rumors about him, like making up stories that he's secretly working for Voldemort again. It could be that these false reports are ruining his reputation and causing problems in his relationships at Hogwarts and in the wizarding community.
Hermione would most likely support Harry. She's always been his loyal friend and would see the injustice of the Daily Prophet's actions. She could help him research laws in the wizarding world (if there are any regarding slander or privacy) and also use her intelligence to find ways to counter the newspaper's claims.
One reason could be defamation. If The Daily Prophet in the fanfiction spreads false and harmful rumors about Harry Potter, that might lead to a lawsuit. For example, if they wrongly accuse him of dark magic practices.
They might have to issue a public apology. This is a common consequence when sued for something like false reporting. It would be a way to try and repair some of the damage done to Harry's reputation.
In this fanfic, Harry could be suing the Ministry over their education policies. For example, Hogwarts might be facing some unfair regulations that are harming the students' learning experience. Harry, having a strong sense of justice, takes on the Ministry. He might gather support from other students, teachers, and even some of the wizarding families who are also affected by these policies. He would then present his case in a wizarding court, which could be full of all sorts of magical procedures and challenges.
Perhaps Harry sues the Weasleys because he believes they misused some of his fame for their own financial gain. For example, they might have used his name to promote their joke shop without his permission. And Harry, being fed up with this exploitation, decides to sue them to set things straight.
Maybe Harry Potter sues the Ministry for false accusations or for not taking proper actions against dark forces. It could also be about some new laws the Ministry made that directly harm Harry or his friends.
Well, perhaps Harry is fed up with the false or malicious stories in the Daily Prophet in the fanfic and decides to take action to destroy it. Maybe he uses magic to burn all the copies or disrupt the printing process.
It could be that the fanfiction used Harry's likeness without permission in a way that was harmful to his image. Maybe it had Harry involved in illegal or immoral activities that he would never do in the original series, so he decides to take legal action against the author of the fanfic.
Perhaps Harry sues the Dursleys for false imprisonment. He realizes that all those years of being confined to his room or the cupboard were illegal. He enlists the help of Hermione, who is well - read in both wizarding and Muggle laws. They present a strong case against the Dursleys. As a result, the Dursleys face public humiliation in their neighborhood, and Harry finally gets the justice he deserves, while also starting to heal from the trauma of his childhood.
Maybe Harry Potter decides to sue the Ministry because they have been withholding important information about his family or his true destiny. It could also be that the Ministry has been unjustly persecuting him or his friends, and he decides to take legal action within the wizarding world's framework.