Honor in that sense isn't easily defined.  It's not the same as the law, it's not the same as morality or ethics.  It may manifest itself in Conduct, it may be said to require certain conduct in certain circumstances as morality or ethics may indicate certain conduct is moral or ethical.  But it isn't itself moral or ethical.  So, I think H.L. Mencken was incorrect when he wrote (taking his phraseology from Nietzsche, with whom he was infatuated) that Honor is simply the morality of superior men

I'd maintain it is a sense of oneself.  That doesn't say much, though, does it?  Perhaps it's better to define it as a self-judgment of one's worth or quality.  Because someone is a person of honor, conduct of a particular kind cannot be countenanced.  There are things an honorable person wouldn't think of doing, not because (or at least not merely because) it would be wrong to do them, but because doing them would be contrary to the nature of a person of honor.  Honor may, therefore, be related to the Stoic injunction that we should act according to (our) nature.

Acting dishonorably would diminish a person of honor.  It would render the person less worthy in that person's eyes, and should in that of others as well, if they're also persons of honor.  It would do so regardless of the consequences of the act or whether the act conforms to law or rules of a particular ethics.

Honor in this sense usually is thought to encompass the possession of certain virtues, most notably honesty, courage, chivalry, justice and nobility.  This may be due to the fact that it appears honor in this sense seems to  have its basis in cultures of aristocracy and rank.   It has been associated with status.  Gentlemen have honor, or nobles do, not merchants or tradesman or peasants.  Honor has been associated with violence as a result, I think.  When one's status is questioned or put in jeopardy, than extreme action may be taken.  If one's honor is impugned, revenge may be called for, or the person being dishonored may have grounds to challenge those questioning their honor, to a duel or seeking some other kind of recompense.  Honor has also been referred to in the military, or at least was in the past, and appears to arise in warrior cultures.

If Honor as commonly defined isn't appropriate except in aristocracies or the military or is an antiquated concept, I think it may survive provided it is seen as a sense of personal worth or merit or status which requires a code of conduct.  A person of honor would act in a particular way without regard to others or circumstances, simply because that is the nature of the person.  The nature of a person of honor would be such that the person's worth, merit or status would be lessened by acting dishonorably, which is to say dishonestly, unjustly, without virtue.  Again, I think the concept of honor so defined would have similarities to what it is that would make a person a Stoic Sage.  I don't know whether Honor as a code of conduct would be a basis for an ethics or morality, but conceiving of it as a part of one's nature seems consistent with the idea that our nature and that of the universe is essentially infused with and devoted to reason, and the belief in an immanent divinity of which we're a part.

Is Honor of any significance, now?  I would say it is not.  Clearly, those who purport to be our leaders have none.  What is significant to them, and to others, is entirely determined by factors unrelated to personal qualities and worthiness.  Generally, those factors are wealth, power, and advantage.  A glance at the headlines is all that's required to establish this is the case.  Politicians are notoriously dishonest, but no person of honor would be able to abide the spectacular, unrepentant dishonesty of the new member of the House from New York without being sickened and ashamed.  Such a person would be shunned by the honorable.  Nonetheless, nothing is being done to  discourage him by the leaders of his party simply because his vote is needed and the potential that there will be one less Republican in Congress outweighs any thoughts of honesty or decency.  A person of honor cannot accept the claim that the end justifies the means, because neither the means nor the end are of importance.  They are, as it were, indifferents.

There is something compelling in the thought that some conduct is unworthy of human beings, something which is more compelling even than the belief that some conduct is wrong, or immoral, or contrary to imagined commands of paternal and paternalistic gods.  Contempt and shame are powerful emotions, particularly when we direct them at ourselves.  Perhaps a sense of honor is something which we should cultivate.