Perhaps in the fanfiction, Snape assigns Harry to clean all the old potion vials in the storage room during detention. While doing so, Harry discovers an old diary that seems to hold secrets about Snape's past at Hogwarts, which then leads to a series of revelations and a change in their relationship.
One interesting scenario could be that during detention, Snape makes Harry clean all the potion bottles in the dungeon while lecturing him about the importance of precision in potions. Harry, in turn, discovers an old, forgotten potion recipe among the bottles and Snape is grudgingly impressed.
Some fanfictions have the idea that while in detention, Harry and Snape are magically bound to work together on a special potion. This potion has the power to reveal hidden secrets. As they work on it, they start to share their own secrets, like Snape's unrequited love for Lily and Harry's fears about Voldemort. It's a really interesting way to develop their relationship.
Well, Harry could be forced to clean the Potions classroom during detention. Snape might make him scrub every cauldron by hand while muttering about how clumsy Gryffindors are. Harry might accidentally spill a potion ingredient, which makes Snape even angrier and gives him more tasks to do, like sorting out all the potion ingredients in the store cupboard.
One possible interesting plot could be Snape having to follow Harry Potter's lead in a dangerous mission. Maybe they are in a situation where Snape has no choice but to trust Harry completely, which is a big change from their usual relationship in the original books.
One interesting plot could be Harry growing up with Snape's strict but ultimately caring guidance. Snape might teach Harry potions from a very young age, and Harry could develop a different kind of magic skill set compared to canon. Another plot could involve Harry having to deal with the Slytherin side of the wizarding world earlier, as Snape has strong Slytherin ties. Also, the relationship between Harry and Snape could be a complex mix of father - son dynamic and teacher - student relationship, which would be really fascinating to explore.
One interesting scenario could be when the characters are in detention and they discover a hidden magical object. For example, in some fanfictions, Harry and his friends might find an old, enchanted diary during detention in the castle. This diary could hold secrets or be a portal to another time or place.
One interesting plot could be Snape raising Harry while secretly training him in advanced potions and dark arts defenses. Harry would grow up with a different perspective on Snape, seeing him as a strict but caring father figure. Another plot might involve Snape having to protect Harry from other dark wizards who suspect Harry's true potential. This could lead to some intense magical battles and Snape using his knowledge and skills to keep Harry safe.
A really unique plot could be Snape adopting Harry during a time when Voldemort is still a threat. Snape has to balance protecting Harry from Voldemort's followers and also from the prejudice within the wizarding community. He might train Harry in Occlumency more intensively, and in the process, they develop a deeper understanding of each other. Harry starts to see Snape not as the mean professor but as a complex person with his own traumas and reasons for his actions.
Well, a possible plot is that Harry discovers his true parentage in a very dramatic way. He could be in the middle of a battle or a difficult situation at Hogwarts and suddenly realize that Snape and Bellatrix are his parents. This would completely change his perception of himself and his place in the wizarding world. He might then have to deal with the consequences of this revelation, such as how the Order of the Phoenix and the Death Eaters view him.
Perhaps in the fanfiction, Harry is magically bound to please Snape in academic tasks. So, he has to study extra hard in Potions class and other subjects Snape is involved in. This might result in a transformation of their relationship, from animosity to a grudging respect as Snape starts to see Harry's efforts.