One common plotline is that Harry's potion - making skills earn him a special place in Snape's class. For example, he might discover a new potion ingredient or brewing method that no one else knew. Another is that his skills lead to him getting involved in some sort of secret or mystery related to potions, like an old potion recipe that holds a key to a hidden treasure.
Often, the plot will involve Harry using his potions skills to help his friends. He could brew a special potion to heal an injured classmate or create a potion that gives them an edge in a Quidditch match. Sometimes, his potions talent is a result of some sort of ancient magic or a hidden family heritage that gets revealed as the story progresses. In many fanfictions, his skills also put him at odds with the more traditional potions - makers who are jealous of his natural ability.
In some 'Harry good at potions' fanfiction, Harry's potions ability is tied to a prophecy or a greater destiny. His unique skills might be the key to defeating a new dark force that has emerged in the wizarding world. The plot may also revolve around Harry using his potions knowledge to reform the potions curriculum at Hogwarts. He might introduce new and more effective ways of teaching potions, which causes a stir among the faculty and students alike.
One interesting plotline could be Harry discovers a hidden potion recipe that gives him the ability to communicate with magical creatures. Another might be Harry has to make a potion to save a sick friend under very difficult circumstances, like with limited ingredients.
One interesting plotline could be the potions professor secretly mentoring Harry in advanced potions outside of class, leading to Harry's rapid improvement and discovery of new potions with unique magical properties.
One interesting plotline could be Neville, who is already good at Herbology, discovers a new way to use plants in potions with Harry's help. This changes the way potions are made at Hogwarts.
Often, they show Harry being confident in the potions classroom, which is a contrast to his original somewhat - nervous self. He might also be more respected by Snape, or at least not constantly picked on.
In a lot of these fanfictions, it could be that Harry has a secret mentor. This could be an old wizard who has been secretly guiding Harry in potions, away from the prying eyes of the Hogwarts staff. This mentor could have seen Harry's potential early on and decided to help him develop his skills in private. Another aspect could be that Harry's own determination and his need to prove himself in a different area other than Quidditch or fighting Voldemort drives him to secretly practice and become good at potions. Also, the idea that there are hidden aspects of magic within him, perhaps related to the founders of Hogwarts, that only manifest in his potions - making ability.
One possible plotline could be Harry finding some ancient potions book in the library that has unique brewing methods. He then secretly practices them in an abandoned classroom.
Yes, there are quite a few. You can check out fanfiction websites like FanFiction.net or Archive of Our Own. Just search for 'Harry is good at potions' in their search bars, and you'll likely find some interesting stories.
One common theme is redemption. Often, the Potions Master is a complex character like Snape, and fanfictions explore his journey towards redemption. Another theme is the exploration of potions as a powerful art. Writers often delve into new and creative potions and their effects. And there's also the theme of mentorship, showing the Potions Master guiding students in the art of potions - making.
Hermione is often a common character. She might be helping Harry with research related to the potions professor's strange behavior or new potions they are working on. For example, she could be scouring the library for information about a rare ingredient the professor has introduced.
One possibility could be a story where Harry discovers a rare potion recipe and masters it, using it to solve difficult problems at Hogwarts.