Well, in a 'new normal' Harry Potter fanfic, perhaps the Ministry of Magic decides to open up more to the Muggle world in order to promote better relations. So, Hogwarts starts an exchange program with some Muggle schools. This would bring a whole new set of challenges and friendships. Also, the way Quidditch is played could change. Maybe there are new safety regulations for players, like wearing special protective charms all the time.
One idea could be that in this new order, Hogwarts starts accepting students from all magical creatures, not just witches and wizards. This would lead to a whole new dynamic in the school, with different cultures and magic types interacting.
One idea could be the new marauders getting into trouble with a new set of magical artifacts that have been hidden in Hogwarts for centuries. They might discover a map that leads to these artifacts, much like the Marauder's Map. Another could be their interactions with the ghosts in Hogwarts in a more adventurous way, like trying to solve a mystery that the ghosts have been keeping for ages.
One idea could be that the new generation discovers a time - turning device left by Hermione. They travel back to the time of the first Triwizard Tournament. There, they meet a young Sirius Black and Remus Lupin, and get involved in the mystery surrounding Cedric Diggory's death in a new way.
Well, you could have a fanfic where Voldemort uses DND - style dark magic to open portals to other dimensions. The Order of the Phoenix has to quickly learn DND - style combat and magic rules to fight off the new threats that come through these portals. Harry could be sent on a special mission to find a DND - style mage who can teach them the counter - spells. Meanwhile, the Death Eaters are also trying to master this new form of magic for their own evil purposes. This would create a really exciting and unique story.
Perhaps a plot where the protagonist can choose to either side with the Ministry of Magic during a time of political unrest within the wizarding world or join a rebel group. This would involve different magical battles and moral dilemmas depending on the choice. In the 'harry potter cyoa fanfic', this could show different aspects of the wizarding society that we don't often see in the original books.
A cool plot for a Harry Potter Twilight fanfic could be a competition. The wizards from Harry Potter and the vampires and werewolves from Twilight compete in a series of challenges. These challenges could test both magic and the unique abilities of the Twilight characters. Or perhaps a love triangle between a Harry Potter character, a Twilight vampire, and a Twilight werewolf. It would be interesting to see how the different cultures and powers interact in matters of the heart.
Another plot could focus on the lives of the secondary characters from the original series. For example, Neville could be the head of Herbology at Hogwarts and discovers a new, dangerous plant that has some sort of connection to an ancient magic. The 'carry on fanfic' could then be about how he and his students deal with it, with some cameos from the main characters.
One idea could be that a dragon escapes from Gringotts and Harry has to use his skills to track it down. Maybe the dragon has a special connection to a powerful dark magic item that Voldemort was after.
One idea could be that Voldemort somehow enters the Runescape world to seek more power. Harry and his friends follow him there to stop him. They have to learn the rules of Runescape quickly to be able to fight Voldemort. Another idea is that a Runescape wizard discovers a portal to the Harry Potter world and brings some of the unique items from Runescape, like special runes, which causes a stir in the wizarding world.
A great plot for a Harry Potter academy fanfic might be a time - travel scenario. A group of students accidentally travel back in time to the founding days of Hogwarts. They have to navigate the different rules and attitudes of that era while also trying not to change history too much. Or perhaps there's a plot where a student is wrongly accused of a major crime at the academy and has to prove their innocence. They'll need to use their wits and the help of their friends to find the real culprit.