Time travel would have a huge impact on Percy's relationships. When he travels to the past, he could meet his ancestors or the predecessors of his friends. This could change how he views his own family and friends in the present. For instance, if he meets an ancestor who was a great hero, he might feel more pressure to live up to that legacy. On the other hand, if he goes to the future and sees how his relationships with his friends end up, it could either strengthen or weaken his current bonds. If he sees a bad future for his relationship with Annabeth, he might try to change things in the present, but this could also lead to unforeseen consequences.
If Percy travels back in time, he might form new relationships with historical characters. For example, he could become friends with an ancient demigod he never met before. However, when he returns to his own time, these relationships might seem like a distant memory to him. With his present - day friends, like Annabeth, time travel could cause some strain. If he changes things in the past, their relationship might be different in the present. Maybe Annabeth wouldn't trust him as much if he comes back having made big changes without her knowledge.
Well, consider Percy traveling back to a time when his father, Poseidon, was young. He could form a different kind of relationship with his father at that time, which would surely affect how he relates to Poseidon in the present. In terms of his friends, if he time travels and gets involved in a past event that changes the fates of some of them, like if he accidentally causes one of his friends to not be born in the present. His relationship with the remaining friends would be filled with guilt and confusion. Also, if he travels to different time periods and meets different versions of his friends, it would be a very complex situation. He'd have to deal with the fact that these are not exactly the friends he knows, but they are also related to them in some way.