To find an example of honesty, take a look at the closing paragraphs of the story. After stealing a horse, Aram's cousin Mourad decides to return the horse to its rightful owner, John Byro.
Although stealing a horse is the very opposite of being honest, the fact that Mourad gives the horse back to John Byro demonstrates that there is indeed a streak of honesty in this family.
In fact, it is worth remembering that Mourad never intended to keep the horse long-term nor to sell it for money. He was simply crazy about horses and was desperate to ride one. It was this desperation which prompted him to take the horse from John Byro. This is, perhaps, evidence that Mourad really is the inheritor of the family's "crazy streak."