He could be more cautious. The perils of Tartarus might have made him less reckless. For example, he might think twice before getting into a fight or taking risks that he used to take without hesitation.
Percy might become more solitary. He has seen things in Tartarus that are beyond the comprehension of most people at school. This could make him feel isolated and less likely to engage in social activities as he used to.
On the other hand, his experience in Tartarus could also make him more empathetic. He has endured so much pain that he might be more in tune with the suffering of others. He could be more likely to reach out to those who are in difficult situations, knowing what it's like to be in a seemingly hopeless place like Tartarus.
It can also lead to new character development arcs for other characters. If Percy is different, those around him have to adapt. Annabeth might have to learn how to deal with this new Percy, which could be a whole new plotline in the fanfiction. Maybe she tries to bring back the old Percy, or she has to fall in love with this new version of him all over again.
In some fanfictions, their relationship could become more complex. They might have seen sides of each other in Tartarus that they didn't expect, and it could cause some friction. But at the same time, they also have this unbreakable connection because of what they've been through. They might argue more about how to deal with the after - effects of Tartarus, like Percy wanting to just forget and move on while Annabeth wants to analyze and learn from it. However, in the end, they still love each other deeply.
Well, they would probably communicate better. In Tartarus, they had to rely on clear communication to survive. So, back in the normal world, this would carry over. They'd be more open about their feelings, fears, and plans, which would strengthen their relationship in a more emotional way.
It makes him more hardened. The harsh conditions force him to become tougher both physically and mentally. He has to constantly be on guard and ready to fight, which sharpens his reflexes and his ability to think quickly in dangerous situations.
His relationships with his friends could become stronger. When he eventually gets out, his friends might see him as even more of a hero for surviving Tartarus alone. They could respect him more and he might be more confident in his relationships.
In some fanfictions, Percy might be more withdrawn.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.
His personality would likely experience a shift towards a more solemn and responsible demeanor. With godly powers come great responsibilities. Percy would no longer be able to act as carelessly as he did as a demigod. He might become more reserved in showing his emotions, especially as he realizes that his actions as a god can have far - reaching consequences. Also, his loyalty might be tested as he has to balance his new divine status with his relationships from his mortal days.
One possible plot could be Percy dealing with PTSD from Tartarus while trying to fit into high school life. He might have nightmares that disrupt his school days and struggle to make friends who can understand his past traumas.
Often, their relationship is shown as deeper and more unbreakable. They've been through Tartarus together, which is like the ultimate test. So in fanfiction, they trust each other completely and are always there for one another.