He might face political intrigue. There are likely those in the Wizengamot who oppose him due to his past actions and his association with Dumbledore. Also, he has to deal with the complex and often archaic laws and traditions of the wizarding world which could be a huge obstacle.
He becomes more politically aware. Through his experiences in the Wizengamot, he starts to understand the power dynamics and the behind - the - scenes workings of the wizarding world. His character evolves from just a hero on the battlefield to a political figure.
Well, in some fanfictions, Harry might gain control over the Wizengamot through his heroic deeds. For example, if he saves the wizarding world from a great threat again, the public support could be so strong that he can influence the Wizengamot's decisions. He could also use his knowledge of magic, especially some ancient or powerful magic that gives him an edge in the political arena of the wizarding world.
Hermione would likely be one of his allies. She's intelligent and has a strong sense of justice, always having been interested in the rights and wrongs of the wizarding laws. She could provide Harry with valuable research and advice.
In this fanfiction, Harry might also form alliances within the Wizengamot. He could team up with some of the more progressive members to counter the conservative forces. His skills as a wizard and his leadership qualities from his time at Hogwarts could come in handy in the political arena of the Wizengamot. He could even start initiatives to improve the relations between the wizarding world and other magical creatures, which would be a significant step forward in the world of Harry Potter fanfiction.
If Harry gets the Potter ring in this fanfiction, it could lead to a whole new adventure. The ring might have a connection to an ancient magic that the wizengamot has been trying to control or suppress. Harry, being the holder of the ring, would be thrust into the middle of this power struggle. He could either choose to work with the wizengamot to manage this ancient magic or go against them if he feels they have malicious intentions.
Obviously, Harry Potter himself is one of the most common. Then there's Dumbledore who often plays a role, either as a supporter or as someone with a complex plan. Hermione is also common as she uses her intelligence to help Harry in the trial.
There are fanfics where the Wizengamot is reformed by Hermione. In these stories, she uses her intelligence and knowledge of the law to make the Wizengamot more just. This often involves her facing a lot of opposition from the old - guard pure - bloods who want to keep things as they are. It makes for a great story as it combines Hermione's character traits with the complex political structure of the wizarding world.
They're popular because the Wizengamot in the original 'Harry Potter' series is often seen as a symbol of the old - fashioned and sometimes unjust wizarding government. Fanfiction writers like to explore the idea of challenging this authority. It gives a sense of empowerment to the characters and readers alike.