In a fanfiction where Percy has a mental disease like depression, it would impact his relationships deeply. His friends would find it hard to understand his mood swings and lack of motivation at times. Annabeth, in particular, would have to be very patient and try different ways to reach out to him. His relationship with his father Poseidon might also be affected as Poseidon might not know how to deal with a son who has such an 'invisible' illness.
Well, let's say Percy has a disease that affects his powers. His relationship with his demigod friends could change. They might start to see him as a liability instead of a strong leader. And his relationship with the gods could also be altered. The gods might either try to use his weakened state for their own ends or completely abandon him, which would have a huge impact on how he views them and his place in the world.
His power can make his relationships more complex. For example, if he uses time control to be with a friend more often, that friend might start to feel like Percy is being overbearing. On the other hand, it could strengthen his relationships. He could save a loved one from a dangerous situation in the past, which would deepen their bond.
With his other demigod comrades, it could change the dynamics in group situations. Some might look at him with pity, which Percy would hate. Others might be inspired by his courage in the face of cancer. It could also lead to some new alliances forming as they all try to find a cure or a way to help Percy manage his condition.
If he has new powers like mind - reading, it might make his friends a bit wary at first. They might be worried he'd pry into their secrets. But over time, it could strengthen their bond as he uses it to understand them better and protect them from hidden threats.
If he gets a new and powerful ability, his friends might feel a bit intimidated at first. For example, if he can control time, they may worry that he could change things without their consent. But over time, they would learn to trust him with this power and it could actually strengthen their bond as they work together to use it for good.
If Percy is taller in fanfiction, it might make him seem more imposing in his relationships. His friends could look up to him literally and figuratively. In romantic relationships, a taller Percy could be seen as more protective. For example, he could easily shield his partner from danger.
It could strain his relationship with Hinata. She would be worried sick about him and might try to get him to rest more, which Naruto, being stubborn, might resist. This could lead to some arguments between them.
His intelligence might make his relationships more complex. With his friends, they could look up to him for his smarts. For example, they might come to him for help with their studies or for solving problems. But it could also create a bit of distance at times. If he's always thinking several steps ahead, his friends might have a hard time keeping up with his ideas.
For other characters, say Nico and Will. If they end up in this scenario, it could be a turning point in their relationship. They might start as just friends, but those seven minutes could make them realize there are deeper feelings there. They could come out with a new understanding and respect for each other, and it could lead to a more romantic relationship later on.
In the fanfiction, Percy's scars could potentially make some people fear him. They might be a sign of the dangerous battles he has been in, and some characters might be wary of getting too close in case they get caught up in similar danger.
His scar might make some people more intimidated by him. They could see it as a sign of his dangerous experiences and be a bit afraid to get too close. For example, new acquaintances in the fanfic might be hesitant to approach him at first.