There could be a really smart professor who suspects there's something different about Percy. Maybe a roommate who is into all kinds of strange and supernatural things without actually knowing the truth about Percy.
There could be a new demigod who transfers to Percy's college. This demigod could have powers related to nature, like being able to make plants grow instantly. They would become Percy's ally in dealing with the campus - based mythological threats. Another new character could be a strict dean who is always on Percy's case, not realizing that Percy is actually protecting the college from real danger. And there might also be a mysterious janitor who actually has a deep connection to the underworld and gives Percy cryptic warnings from time to time.
One potential new character could be a scientist, Dr. Thompson, who is studying the strange phenomena in Alaska that are actually related to the magical world Percy is part of. He initially doesn't believe in the magic but as he witnesses Percy's powers, he becomes an ally. There could also be a group of Inuit shamans who have their own powers and traditions. They might initially be wary of Percy but then come to respect him and work together with him to solve a problem that is endangering both the magical and non - magical parts of Alaska.
Maybe Percy will face new challenges in college like dealing with difficult classes. He could also meet new friends or enemies. And perhaps he'll discover new powers or secrets related to his special nature.
There could be a strict Roman professor who is very traditional and gives Percy and Annabeth a hard time in their classes. This professor might not trust the Greek demigods at first and constantly tests them. Another potential character could be a Roman demigod who becomes their friend. This demigod could introduce them to all the cool spots in New Rome and help them understand the Roman way of life better.
There could be a new bully at Goode who targets Percy because he senses Percy is different. This bully might turn out to have some connection to the mythological world, like being a descendant of a minor god.
The plot could center around Percy's struggle to fit in at college while still being a hero. He may find that the other students don't understand his need to disappear suddenly when there's a monster attack. He could also get involved in some sort of college - wide mystery that has roots in the ancient world. Like, strange symbols start appearing all over campus that only he can decipher because of his knowledge of Greek and Roman mythology. And he has to team up with some unlikely allies, like a nerdy professor who turns out to have a hidden knowledge of the demigod world, to solve the mystery and save the college from an impending doom.
Percy Jackson himself is of course very popular. His friends like Annabeth Chase also often appear. Grover could be there too, still being Percy's loyal friend. New characters might include college professors who turn out to have some connection to the world of mythology, like a history professor who is secretly a descendant of a minor god.
Obviously, Percy would be the central character. His friends like Annabeth, Grover, and other demigods from Camp Half - Blood might also be important. They could be trying to help Percy or might be affected by his insanity. For instance, Annabeth might be desperately searching for a way to bring Percy back to his senses.
Some of the Roman gods or goddesses might also play important roles as characters. They could interact with Percy in different ways compared to how they interact with the Roman demigods in the camp. For instance, if Jupiter shows up, he might have different expectations of Percy compared to Poseidon, Percy's father in Greek mythology.
He would surely interact with Annabeth. Their relationship is central in the Percy Jackson series, so in a Tartarus fanfic, she'd likely be there to support him or be in a similar situation as him.