Often, Harry Potter himself is a main character. Since it's related to the Harry Potter universe, he is likely to be involved in the story centered around the millenium puzzle.
One interesting idea could be a new magical creature introduced through the millenium puzzle. It might be a creature that holds the key to an ancient mystery in the Harry Potter world.
The Marauders, James Potter, Sirius Black, Remus Lupin, and Peter Pettigrew, are still popular in Harry Potter millenium fanfiction. Writers often explore their time at Hogwarts in the past within the context of how it affects the millennium. For example, a story could be about a hidden secret they left behind that resurfaces in the new millennium. Albus Dumbledore also remains a prominent figure in many fanfictions. His wisdom and the mystery surrounding his past are often delved into, especially in relation to the changes and events that occur in the millennium within the wizarding world.
Well, one plot twist could be that Naruto's ability to solve the Millenium Puzzle unlocks a hidden power within him that's different from the Nine - Tails chakra. Maybe it gives him the power to time - travel, and he has to use this new power to prevent a great disaster in the ninja world that was foreseen by the ancient spirits connected to the Millenium Puzzle.
One interesting 'Harry Potter millenium fanfiction' story could be about how the wizarding world adapts to the new millennium. For example, they might have to deal with new forms of magic that are emerging due to the change in the era. Another could be a story where Harry Potter's children face a new threat that is related to the millennium and the ancient prophecies. There could also be fanfictions that explore how Hogwarts itself changes over time with the new millennium, like new courses being introduced or new areas of the castle being discovered.
His relationship with Kakashi could also shift. Kakashi, being experienced, would be both intrigued and worried about Naruto having the Millenium Puzzle. He would try to guide Naruto in understanding and using it properly. Naruto would respect Kakashi's wisdom more in this scenario and would look to him for advice on how to handle the power and the responsibilities that come with the Millenium Puzzle.
They could use magic spells to calm the dragon first. For example, the 'Diffindo' spell might be used to cut ropes that are binding the dragon in an aggressive way, showing the dragon that they mean no harm. Then, they can start communicating with it using Parseltongue if they have that ability, like Harry does. Maybe the dragon has some knowledge about the puzzle and can give hints if it trusts the characters.
One idea could be a dragon guarding a mysterious puzzle that holds the key to an ancient magic. The main characters, like Harry and his friends, have to solve the puzzle while also taming the dragon. Maybe the puzzle is made of dragon scales and only a true Gryffindor can figure it out.
One idea could be a story where Carbuncle gets lost in a mysterious new realm within the Puzzle and Dragons world. It has to use its wits and unique abilities to find its way back. Maybe it teams up with some unlikely allies along the way.
It's not a common type. In the vast world of Harry Potter fanfiction, you'll find a lot more stories about Hogwarts during different eras, the relationships between the main trio, or the exploration of the wizarding world's politics. 'Breeder fanfiction' is more of a specialized sub - genre that only a small subset of fans might be interested in, as it delves into family - building and genetic - like concepts which are not as central to the general appeal of the Harry Potter series.