His friends like Annabeth or Grover could be the ones to rescue him. They might discover a way into Tartarus and brave all the dangers to get to Percy.
It could be an unexpected character. Maybe a minor god who has some connection to Tartarus and feels sympathetic towards Percy. This god could use their unique powers to enter Tartarus and bring Percy out, all while facing opposition from the other forces in Tartarus.
In a 'Percy thrown into Tartarus' fanfic, Percy could actually rescue himself. He could find some sort of power - up or ancient knowledge within Tartarus that allows him to break free. He might have to fight his way through the guardians of Tartarus and use his new - found abilities to make his way back to the world above.
One possible plot could be Percy's struggle to survive in Tartarus. He might have to face the various terrifying monsters there, like the giants or the powerful demons. Maybe he'd find some unexpected allies among the damned souls, and together they'd try to find a way out.
The Titans that were banished to Tartarus could be major antagonists. They might see Percy as an intruder and a threat, and try to stop him from escaping or even try to use him for their own purposes. Also, the various monstrous creatures native to Tartarus, like the Empousai or the Arai, could constantly be attacking Percy.
Annabeth could be a likely candidate. She's always been by Percy's side, and she'd stop at nothing to save him. Her intelligence and combat skills would be useful in the dangerous environment of Tartarus.
It's possible that an unexpected character like a friendly Titan could rescue Percy. There could be a Titan who has had a change of heart and realizes that Percy is important for the balance of the world. This Titan might use its own knowledge of Tartarus and its powers to free Percy from the clutches of that terrifying place.
The Titans or Giants that were banished to Tartarus could be the main antagonists. They might see Percy as a threat or an opportunity to escape and seek revenge on the Olympians.
Survival is a big one. Tartarus is a very hostile place, so Percy has to fight to stay alive. Another theme is redemption in some cases, like if he feels responsible for getting thrown in there.
One popular 'percy thrown in tartarus fanfiction' is 'Into the Abyss'. It really delves into Percy's struggles and his attempts to survive in Tartarus. The author does a great job of describing the terrifying environment.
One possible plotline could be Percy's struggle for survival in Tartarus. He might have to face off against various terrifying monsters. Another could be his journey to find a way out while dealing with the psychological trauma of being in such a hellish place. And perhaps he discovers some long - hidden secrets about Tartarus itself during his time there.
Maybe an old and powerful monster that has long been trapped in Tartarus could be the main antagonist. It sees Percy as a threat or a source of entertainment, and it tries to stop him from finding a way out or from getting stronger.
Perhaps one of the Titans who were banished to Tartarus. They might see Percy as a threat to their plans of revenge or escape. Kronos, if he still has some form of existence in Tartarus, could be a major antagonist. He would have his own motives and powers, and he could try to manipulate Percy or simply get rid of him.