Why did the boys return the horse back to the owner?

One of the boys, Mourad, has been for some time in the habit of taking the horse for early morning rides and returning it before its absence is noted. He never intends not to return the horse. After the horse runs away, however, meaning that the boys can't return it in the morning as intended, they decide to keep it for a few more months so that Aram can learn to ride. Then, a few weeks later, they have a chance encounter with the horse's owner, John Byro. During this encounter, Byro seems to recognize the horse as his own. He inspects the horse's mane and its mouth, and says, "I could swear it is the horse that was stolen from me," and "I would swear it is my horse if I didn't know your parents." The boys then return the horse the next morning to Jon Byro's vineyard.
Earlier in the story, Aram, the narrator, emphasizes how important reputations are in this small community. He says that his own family, and, therefore, that of his cousin, Mourad, have been "famous for (their) honesty for something like eleven centuries." Byro also emphasizes this point further when he says, to the boys, "The fame of your family for honesty is well known to me." The main reason why the boys return the horse is to protect the reputation of their family. It is likely that Byro knows very well that the horse is the same horse that has been stolen from him, but he is giving the boys a chance to return the horse of their own volition. When he reminds them of the reputation of their family, he is doing so as a gentle, veiled threat.

The boys return the horse to the barn of its owner, John Byro, because they recognize that he knows they have it. Having visited their house already to inquire about the horse John Byro then encounters the boys on the road with the horse and asks to look into its mouth. Through a survey of the teeth, he is able to establish that it is his horse, but he is careful in what he says to the boys--he tells them that it is "the twin" of his horse, and if he did not know their parents so well and know them to be an honest family, he would think it was the same horse. Presumably, this trick works on the boys, who then feel guilty and rather ashamed; their household is known for its honesty and the boys don't wish to steal, nor do they want John Byro to tell their parents that they have taken his horse and do not intend to return it. As such, they take the horse back immediately, rather than keeping it for another few months as they had intended.