Perhaps the fanfic focuses on Harry's emotional journey as he unleashes his power. He could start off being afraid of this newfound strength, but then learns to embrace it. He might face some internal struggles, like whether using his power is the right thing to do, especially if it has some unforeseen consequences.
Obviously, Harry Potter himself would be a central character. His friends like Hermione and Ron would most likely be there too. They could play a role in helping Harry understand and control his power, or they might be in danger and Harry's unleashed power is what saves them.
In the 'harry shows his power fanfic', perhaps it's a story where Harry has to show his power in a wizarding competition. He might use his unique abilities like Parseltongue to outwit his opponents and prove his worth. He could also draw on his connection with the Patronus charm to protect himself and others during the tasks.
Well, it could be about a new dark wizard emerging in the wizarding world and the new generation of Hogwarts students having to rise to the challenge, much like Harry and his friends did before. Maybe they discover a new kind of magic that is related to this power's awakening.
Perhaps it's a story set in a fictional world. Harry and Lemon could have special powers. They use their powers to help their community or to solve a mystery that has been plaguing their town. They might start as normal siblings but through their experiences, they grow and become more confident in themselves and their abilities.
Maybe Harry hides his intelligence to avoid unwanted attention at Hogwarts. He could be tired of being singled out or feared because of his smartness.
Maybe in the fanfic, Harry loses his temper when he discovers someone has betrayed his trust, like a false friend who leaked his secrets.
Well, it could be that Harry meets his parents through some magical time - turner accident. Maybe he's on a mission in the past and stumbles upon them.
In a 'harry thinks his name is freak' fanfic, it could start with Harry being in a new magical environment, like a hidden part of the wizarding world. Here, everyone refers to him as 'freak' for some reason he can't understand at first. It could be a cultural difference or a misinterpretation of his powers. As he tries to adapt to this new identity, he discovers secrets about this part of the wizarding world that are related to his real name and his true place in the magical community.
Maybe Harry realizes he made some mistakes in his past and decides to go back in time to correct them. He could try to save his loved ones who died in the original story.
Perhaps the cloak is a family heirloom that the boy only discovers when he's in a time of great need. It could be a cloak that was enchanted by an ancestor with a specific purpose, like protecting the wearer from a particular dark magic. As he starts using the cloak, he realizes that it comes with a set of rules and responsibilities. He has to figure out how to use it to his advantage while also fulfilling whatever the cloak's original purpose was. This could lead to him learning more about his family's past and his own place in the wizarding world.