Perhaps a prophecy misinterpretation. If a prophecy was misread and it seemed like Percy was the cause of great evil to come, the gods might take drastic action. For instance, if the prophecy spoke of a 'child of the sea' causing destruction, and they wrongly assumed it was Percy without fully understanding the context.
Well, one reason could be that he's wrongly accused of a major crime against the gods. Maybe some false evidence is planted, making the gods believe he's a threat, so they banish him. Another reason might be that he's trying to protect someone or something, and his actions are misinterpreted as going against the gods' will. For example, if he's trying to save a powerful but 'forbidden' creature, the gods might see this as a violation and send him to Tartarus.
Well, perhaps in the fanfic, there's a prophecy that gets misinterpreted. The gods think Percy is going to cause great harm based on this wrong interpretation, so they send him to Tartarus. It could also be that there's some sort of power struggle among the gods. Percy might be caught in the middle of it, and one side decides to get rid of him by banishing him to Tartarus to tip the balance in their favor.
A third plotline could be about Percy's mental and emotional state during his banishment. He could be dealing with feelings of betrayal, as he was banished. He might question his identity and purpose, and throughout the fanfiction, he gradually overcomes these doubts and insecurities, emerging stronger in character even if he's still trapped in Tartarus.
He might rely on his powers. Percy is a son of Poseidon, so his control over water could be crucial. He could use water sources in Tartarus, if any, to heal himself, create barriers, or even find a way to escape. For example, he could manipulate the Styx water to his advantage.
Maybe it's part of a plot to have Percy gain some sort of power or knowledge. In many fictional stories, a character has to go through great pain and suffering before emerging stronger or wiser. The torture in Tartarus could be a way to set up this transformation for Percy.
He might rely on his skills and the powers he already has. His control over water could help him find sources of water in Tartarus for survival. Also, his combat skills would be crucial in fending off the various monsters there.
In some fanfictions, Percy might be banished because he goes against the traditional ways of the demigod society too much. For example, if he tries to form an alliance with a group that the other demigods consider enemies. His actions could be seen as a threat to the stability of their world, so they banish him to keep the peace.
Maybe he accidentally breaks some ancient and unforgivable rule. For example, he could have uncovered a secret that the gods wanted to keep hidden and in their attempt to silence him, they banish him. In fanfiction, the writers often play with the idea of Percy's curiosity getting him into big trouble with the Olympians.
Well, one possible plot could be that Percy and Zoe have to face various ancient and terrifying monsters in Tartarus. They might encounter challenges that test their friendship and their individual strengths. For example, they could come across the Titans that were once banished there, and have to find a way to avoid or defeat them while also trying to find a way back to the upper world.
In a 'percy banished to tartarus fanfic', the plot could be about Percy's internal battle as well. He's been banished, which could make him feel betrayed and angry. So he not only has to deal with the physical threats in Tartarus but also his own emotions. He could question his identity and his purpose, and this self - discovery could be a major part of the story. He might start to see things from a different perspective, and this new understanding could be crucial in his journey to either escape or change his situation in Tartarus.