Let's be honest with ourselves.
Percy didn't reject godhood for Annabeth. Even if he was thinking of her in that moment, he was well aware of gods marrying mortals and ascending them. He also had no reason to think Annabeth wouldn't want that.
A part of it was definitely using his gift to make changes for demigods, but he also could have done that as a god. Hermes said it himself: the gods would most likely forget about the promise they made Percy. Percy also knew they did not stick to their oaths well. He and Thalia were proof of that. He would have made a bigger impact by being a god and pushing for change and being there to make sure they stuck to it.
No, Percy didn't want godhood because he didn't want to live forever. We see it the first time he meets Dionysus and Chiron. He thinks that he should think it’s cool, but even at twelve, the thought of living forever is daunting and dreadful.
Furthermore, Percy is frequently actively suicidal. He thinks about it at least three times between the PJO and HoO arcs. Even without those specific instances, Percy was fully prepared to die when he was 16 and clearly doesn’t know what to do with himself afterward. Everything he does is purely what everyone else expects him to do, Annabeth especially.
He doesn’t go on quests when Apollo is mortal because his mom would stress too much during the pregnancy. He decides to finish school and try to get into NRU to be with Annabeth. Even taking on quests again to do it, because he’s convinced that a picket-fence life will make him happy and he believes New Rome will be the only way he can do that.
But, he only thinks that’s what will make him happy because it’s the one thing he’s never had. Percy would go nuts sitting in college classes, being surrounded by stuck-up pedigree Romans who shit on him and Annabeth because of their parents. He’d go stir crazy after a few months in New Rome. Kid couldn’t even make it the whole summer at Camp Half-Blood.
So, he tries to make it work thinking it’ll make him happy because the truth is:
Percy didn’t want to be a god because he wants to die.
