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.
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.
Maybe he wants to keep up with the latest news in the wizarding world. There could be important announcements, like news about Voldemort's activities or updates on the Ministry of Magic.
He might do it because the Daily Prophet has been printing untrue stories about him.
He might be interested in the articles about new magical creatures. Since he has had many encounters with different and often dangerous creatures, news about newly discovered ones or updates on known ones would surely catch his attention.
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.