Survival is a big one. Tartarus is such a harsh and dangerous place, so Percy has to find ways to stay alive. Another theme is self - discovery. Facing the extreme challenges there makes Percy learn more about his own strength and limitations.
One great 'Percy jumps into Tartarus' fanfiction is 'Into the Abyss'. It really delves deep into Percy's psyche as he faces the terrors of Tartarus. The author does an amazing job of describing the dark and desolate landscape, and how Percy has to use his wits and powers to survive.
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.
In some fanfictions, Percy might be more withdrawn. He has seen the horrors of Tartarus and it could make him less trusting of the world around him. He may also have new powers or a deeper understanding of his abilities that he gained from the experiences there.
Well, one aspect to consider in this fanfiction could be the character development. Percy Jackson is already a well - established character, but as the heir of Tartarus, he might face new challenges that force him to grow in different ways. Maybe he has to deal with darker powers within himself. Another point could be the world - building. Tartarus is a very dark and dangerous place in the Percy Jackson universe, so how the author expands on that and shows its influence on Percy and the story is important. Also, the relationships with other characters would be interesting. Will his friends still trust him when they find out he has this new 'heir' status? How will his enemies react?
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.
One possibility could be Percy's struggle to survive the harsh conditions. He might have to find new ways to use his powers in Tartarus. For example, the strange and dark magic there could either enhance or limit his abilities. He could also form alliances with unexpected creatures. Maybe he teams up with some of the ancient and terrifying monsters that live there, but they have a common enemy, so they work together for a time.
Well, one possibility could be Percy facing his inner demons. In Tartarus alone, he might have to deal with the memories of his past failures and the fear that comes with being in such a desolate place. He could encounter strange and terrifying creatures that test his combat skills and his will to survive.
In some fanfictions, Percy might gain power in Tartarus through facing and overcoming numerous deadly challenges. For example, by defeating powerful monsters that inhabit Tartarus, he could absorb some of their essence or gain knowledge from them which then transforms into power. Another way could be through making pacts with certain entities in Tartarus in exchange for power, although this would likely come with a price.
Well, in many fanfictions, Percy might be on a mission to assassinate some powerful being in Tartarus. Maybe he has to face all kinds of hellish creatures and overcome insane challenges. It could involve him using his water powers in unique ways, like creating weapons out of water to fight the demons of Tartarus.