His character could develop in a darker direction. The Venom symbiote often brings out the worst in its hosts. So Harry might become more aggressive and self - centered. He could start to view his friends as obstacles and become more isolated.
One idea could be that Harry Osborn, in his search for power to overcome his family's legacy of misfortune, somehow comes into contact with the Venom symbiote. Maybe he initially tries to control it for his own gain, like using it to take over Oscorp. Another plot could involve Venom influencing Harry's darker side, causing him to clash with Spider - Man in new and unexpected ways.
One great fanfiction idea could be Harry Osborn's internal struggle with his new - found powers in 'The Amazing Spider - Man 2'. He might be torn between using them for good, like Spider - Man, or for his own self - interests. For example, he could be constantly fighting the urge to use his powers to gain more control over Oscorp, while also being aware that Spider - Man's way of using powers is more noble.
One idea could be a story where Peter tries to protect Harry from the darker aspects of their world. Since Peter has his Spider - Man powers, he could be constantly on guard to keep Harry safe, perhaps from the enemies he made as Spider - Man. For example, a villain might target Harry to get to Peter.
You can start by looking on fanfiction websites like Archive of Our Own (AO3). It has a large collection of fanfictions, and you can easily search for Peter Parker and Harry Osborn stories using the search bar. Just type in their names and you'll get a list of relevant fanfictions.
There's also 'Friendship Forged in Crisis' which is an amazing Peter Parker and Harry Osborn fanfiction. It focuses on a major crisis in the city where they have to rely on each other completely. The story has a lot of action and also some really touching moments between the two characters that really bring out the essence of their friendship.
A fanfiction could focus on Harry Osborn's life at school. He's rich and popular, but in this version, he's also secretly dealing with family issues related to his father's business. He could cross paths with Peter more often at school, and their interactions could be a mix of rivalry and a budding friendship.
I like the storylines where Harry tries to break free from the Goblin curse. It shows his internal struggle and his will to be a better person. It's a powerful concept that adds depth to his character.