honestly i think the reason as to why percy is so hard to place in one hogwarts house is because of the fact that he displays such strong qualities that overlap between the houses. if we look at the qualities that percy possesses compared to the characteristics stated in the sorting hat’s song, he embodies almost all of them.
foremost, in the gryffindor verse it says that gryffindors are brave at heart, and are daring, have great nerve, and are chivalrous. We have seen throughout many books that he is indeed brave and daring, and he does have nerves of steel at some points (ex, jumping off the st. louis arch and falling with annabeth into tartarus). He also shows great acts of chivalry & this aspect is also seen throughout other parts of his personality
as for the hufflepuff verse, it says that members of the house are just and loyal, as well as being patient and unafraid of toil. clearly, percy displays a lot of loyalty throughout the books, whether it be for annabeth or his mom or for his camp. he is also not afraid of fighting anyone or anything in order to protect those he is loyal to.
in the ravenclaw verse it is less obvious that percy shares some similarities to that house (seeing as it talks about having a ready mind, having a lot of wit & will to learn) however, percy does display these qualities throughout both series, as he wants to learn any different things (ex. learning how to defeat kronos, learning how to unite both the greeks and romans, learning how to change the way he perceives the world). also he shows how witty he is through both his sense of humor and how fast his mind works in different situations (mostly fights, etc)
now the slytherin verse talks mostly about finding real friends, and calls slytherins cunning and ambitious. percy obviously cares a lot about finding his real friends (and he finds them in annabeth, grover, the seven (eventually), etc) so that’s a pretty obvious one. however percy is exceptionally cunning, whether it be in battle or just everyday life. correct me if im wrong, but im pretty sure annabeth in the books has said how intelligent percy is (even though he comes off as stubborn or oblivious at times)
in all honestly im not sure what kind of conclusion i was trying to make by writing all of this out, but the one i have come to is that all of these characteristics, although make him hard to classify in one house, show that percy is an exceptionally strong protagonist that exhibits powerful qualities that make him a hero.
Therefore, i am concluding that this whole sorting thing is kind of backwards because percy doesn’t really fit in any of the houses because he has shown that he belongs in all of them