Friendship or a deeper relationship development is also common. Since they are from the same anime universe, writers often explore how they might become closer, whether it's through shared battles or just spending time together. Maybe they start off as rivals and slowly become friends or something more.
You can try searching on fanfiction websites like Archive of Our Own or FanFiction.net. Just type in 'Katekyo Hitman Reborn Dino x Hibari' in their search bars and you might find some interesting stories.
One idea for an OC in Katekyo Hitman Reborn could be a gamer who has the ability to use video game skills in the real world of the series. For example, they might have a 'health bar' that others can see, and they can level up their combat skills like in a game. This would add an interesting dynamic to the story as they interact with the existing characters.
There could be a fanfic that focuses on a romantic relationship between a character from Katekyo Hitman Reborn and Sakura. Let's say Hibari falls for Sakura when she stumbles into his territory. He is initially hostile but then gradually becomes interested in her unique charm, and the story could explore how their relationship develops against the backdrop of the Reborn world.
The Hitman Reborn manga ended with a climactic battle and resolution of the main character's story arcs. But it's a complex ending that's best discovered by reading it yourself.
One interesting 'hitman reborn fanfic' could be about Tsuna's further adventures in the mafia world that the original series didn't fully explore. For example, a story where he has to face a new and powerful enemy alliance all by himself without the immediate help of his guardians.
Begin with the stories recommended by other fans. There are often forums and groups dedicated to Katekyo Hitman Reborn where fans discuss their favorite fanfictions. These recommendations can lead you to some really great reads. Also, consider looking for stories that focus on your favorite characters from the series. For example, if you like Tsuna, look for fanfictions that center around him.