Having got a name for his horse so much to his taste, he was anxious to get one for himself, and he was eight days more pondering over this point,
I believe that the choosing of the horse's name and the horse's name itself are important actions that happen in this piece. They are notable because the name "Rocinante" means "ranked before all other horses." Don Quixote believed that he was ranked above everyone else and that his horse deserved a name that reflected that as well. Details like this give subtle hints to what Don Quixote's personality might be like.