The most honourable way to lose? When in a bad position, is it more honourable to resign or to face defeat? And if it is to face defeat then is there a time when resigning is considered an honourable thing to do, or is it always considered the cowards way out?
Playing on in a hopeless position when it is clear to you that your opponent has the knowledge, skill and time to finish you off is never honourable. It shows a lack of respect.

If you have played your moves much faster than your opponent so that he is in serious time trouble and you aren't, then it is perfectly reasonable to play on. With just a few seconds left playing without increment he may not have enough time to deliver the coup de grace.

In the reverse situation where you are the one with very little time on the clock it is excusable but perhaps not particularly honourable to play on as you can always claim that with so little time to think the hopelessness of your position hadn't sunk in.

Note that positions where you have very few moves and so stalemate is a possibility are not hopeless. Nor are positions where, although material down and so expected to lose with best play, you have some prospects of counterplay in a double-edged position.