One common plotline is them working together on a project. For instance, they might be part of a team to reform the Ministry's education policies. Percy brings his administrative knowledge and Hermione her experience as a student and her innovative ideas.
One common plotline could be that Hermione is betrayed by her friends in Hogwarts. For example, they might believe false rumors about her and turn their backs on her. Another could be a romantic betrayal, like if Ron, in a different take on the story, cheats on her or breaks her heart in some way. Sometimes, she could be betrayed by the wizarding community as a whole, perhaps being wrongly accused of dark magic and no one believing in her innocence.
Often, in these fanfictions, Hermione is used as a tool to understand Voldemort better. There might be a plot where she is forced to enter his mind, like in a Legilimency - related situation. This could create a connection between them that the fanfiction then explores. Another common plot is that Hermione's magic has some sort of resonance with Voldemort's, which draws them together in a relationship that is both dangerous and full of potential for new storylines.
One common plotline is that Hermione and Snape are forced to team up to deal with a werewolf outbreak. Maybe the werewolves are attacking Hogwarts students, and they have to put aside their differences to protect the school. Another is that Snape has some past connection to werewolves that Hermione discovers, which changes their relationship. For example, he might have been involved in some secret research about werewolves in the past.
Often, there's a plotline where Hermione's epilepsy affects her performance in school. For example, she might have a seizure during an important exam. But then, this event could lead to her classmates realizing the importance of support and acceptance. Also, there could be a plotline where she meets a new character, perhaps a healer or an old wizard, who has some unique knowledge about dealing with her condition.
One common plotline could be Hermione using her skills as a trained soldier to defend Hogwarts. She might lead a group of students in a battle against dark forces, using her strategic knowledge. Another could be her being sent on secret missions outside of Hogwarts to gather intelligence or rescue captured allies. For example, she could use her intelligence and combat training to infiltrate a Death Eater stronghold.
One common plotline is Hermione being captured by Bellatrix and forced into a situation where she has to act like a pet. For instance, Bellatrix might use a spell to make Hermione obey her every command.
Sometimes, they get injured accidentally, like in a potions accident gone wrong in the Hogwarts dungeons. This then leads to a series of events where they learn more about each other's vulnerabilities and strengths. It can also bring out the protective sides of their friends, like Harry trying everything he can to make sure they are okay.
One common plotline is the adjustment period. Hermione and her husbands have to get used to living together and sharing their lives. Another is dealing with the outside world's reactions. Since it's an unusual situation in the wizarding world, they might face criticism or curiosity from others. There could also be a plotline about how they divide household and magical responsibilities among them.
In some fanfictions, Artemis may hurt Percy because of her strict sense of justice. Say Percy breaks one of her rules unknowingly, like entering a sacred forest she has marked off. Artemis, being the goddess of the hunt and protector of the wild, might feel the need to punish him. This could lead to Percy getting physically injured, like being struck by one of her arrows. Or maybe Artemis is under a curse that makes her act out of character and she ends up hurting Percy without really meaning to. This then sets off a chain of events in the fanfiction where Percy has to deal with the injury and also try to find a way to break the curse on Artemis.
One common plotline could be Hera trying to manipulate Percy's relationships with others. For example, she might try to keep him away from Annabeth to assert her possessiveness. Another could be Hera interfering in Percy's quests, not to help him but to keep him under her control.