A very common plot might be centered around loyalty. Harry would need to ensure the loyalty of his followers, just like a mafia boss. There could be betrayals within his group that he has to deal with. Also, he might be involved in some sort of trade, perhaps of illegal magical potions or artifacts. And he would be constantly trying to outwit the authorities in the wizarding world who are trying to bring him down.
One common plot could be power struggles. Harry Potter, as the mafia boss, might be constantly fighting other powerful wizards or groups who want to take over his 'territory' in the wizarding underworld. Another plot could be about revenge. Maybe someone from his past in the wizarding world wronged him, and now he's using his mafia - like power to get back at them.
Well, in such a fanfiction, it could be that Harry Potter uses his magical skills in a more devious way like a mafia boss. He might use spells to intimidate or protect his 'territory' in the wizarding underworld. For example, he could use the Imperius Curse in a more mafia - like control way over other wizards to do his bidding.
Power is a common theme. Harry, as a mafia boss, is often depicted as seeking and wielding power in different ways compared to the original series. Another theme is loyalty, as in the mafia world, loyalty to the boss and the group is crucial.
One could imagine that a distant relative of the Black family, who has been disgraced or cast out, becomes a mafia boss. They still have some of the Black family's wealth and dark magic knowledge, but use it in a more criminal way. For example, they could be running a protection racket in Knockturn Alley, threatening shop owners to pay up or face dark magic. Severus Snape also could potentially be a mafia boss in an alternate story. He has the intelligence and the ability to be secretive. He could be using his potions expertise to create and sell illegal or dangerous potions on the black market, while also maintaining a network of spies and henchmen within Hogwarts and the wizarding world.
One possible plotline could be Harry rising to power within the wizarding mafia world. He might start from a small - time operator and gradually build his empire. Another could be his conflicts with other powerful wizarding families who are also involved in mafia - like activities. He would have to use his magic and wits to outmaneuver them.
Common themes include conflict. There will be conflict between the values of the mafia, like violence and control, and the values of the wizarding world such as justice and equality. Redemption can also be a theme, where perhaps the mafia boss has a chance to change his ways in the magical context. Additionally, there is often the theme of discovery, like the mafia boss discovering the existence of magic or a wizard finding out about the mafia underworld.
Another theme is secrecy. Harry usually has to keep her double life as a mafia boss and a wizard (if it's set in the HP universe) a secret. So, there's a lot of sneaking around and hiding her true identity from different parties.
There are also plots where different dimensions are on the verge of colliding. This creates a sense of danger and urgency as the characters have to prevent the collision which could have catastrophic consequences for both their world and the other dimensions. In some fanfictions, a villain might be trying to use the power of different dimensions to take over the wizarding world.
One plot idea could be that the mafia boss is secretly protecting Hogwarts from a new dark threat. He uses his underworld connections to gather information that the Ministry of Magic can't access. For example, he might discover a group planning to steal powerful magical artifacts from the school. Another idea is that the mafia boss is actually a wizard who was exiled from the wizarding world long ago and has built his own criminal empire in the Muggle world. But when he hears about Voldemort's return, he decides to use his resources to help Harry Potter, in his own unorthodox way. A third idea could be that the mafia boss has a vendetta against a Death Eater who once wronged him in the past, and he teams up with Harry and his friends to take the Death Eater down.
Power struggles are a common theme. Just like in real mafia stories, in 'Harry Potter mafia family fanfiction', families or groups fight for control over certain areas of the wizarding world, be it magical resources or influence. Another theme is secrecy. These mafia - like families often have many secrets that they are desperate to keep hidden from the rest of the wizarding community.
A common plot is the expansion of Dumbledore's Army. Writers often explore new members joining, different training methods, and more elaborate secret operations against Umbridge. Another plot is the personal vendettas students have against Umbridge. For example, Harry might be more directly confrontational in fanfiction compared to the books.