Assuming timing and accuracy, what are or what is the most important elements of a punch?

I would have to say intend. A lot of punches turn out half@$$ed just because the intend is not there. ...and then those things that Possum added later of course.

If you are not accurate, the punch is ineffective.

If you are late - or too early - the punch is ineffective.

Inaccuracy and ill timing are tels: they say something about your experience, maybe fear, maybe injuries, maybe your stamina. Further, depending on how they land, they can further enrage your opponent.

I can't see how one is more important than the other: I say they are equally important. And critical.

EDIT: Sorry Jim, I thought you wanted to assume only these factors.

In that case, there is the shape of the hand. Foul it up, and you injure yourself.

Then there is the body mechanics (hip rotation, stance). Foul it up, and the punch is ineffective.

Then there is the target - which I distinguish from accuracy. If you hit the wrong target (but properly aim for the wrong target), your punch is ineffective.

I think I might trade "injury to myself" for "ineffective punch". An injury to myself prevents me from "living today, fight tomorrow", so to speak. Therefore, I place "shape of the hand" above the other factors. This includes closed fist, and proper wrist alignment.

I tend to focus on this in my teaching, by creating drills that focus on the hand shape. Specialized exercises for hand shape and wrist alignment include clenching a fist with a pencil, or doing knuckle pushups.

I don't think the other elements are unimportant, but if I had to choose something that is more important, then this is what I would choose: hand/wrist shape/alignment.

EDIT: I agree with Pugpaws - I don't think there is a right or wrong answer, it can be endlessly debated. That in of itself allows to share knowledge and insight. But you ask a fair question, so, I will vote - as I did on hand shape/alignment.

My main instructor used to ask the students which of the following is most important...?

Power, Speed, Timing, Position, Accuracy. All are important, but one is the most important. If it is not right at the instant you act it others are of little use to you. The answer we could debate forever and some people would still not agree. However, we always say Position is the most important because if it is not right at the instant you need to act, the others don't matter.

Sorry if this is not in line with your question. Just thought it relevant to the subject in general.

...

The intent of the punch, the target, power, speed, and distance. this is not in any certain order as all are important.

If you punch with no intent that that is useless. You must have a target. We are not just punching air. You must have power distance and speed as these are all part of having good technique. You already mentioned timing and acuracy.

There is also focus and intent t consider besides those listed by Pugpaws and possum. There are arguments that can be made to support all aspects. Any chain is only as strong as the weakest link, the same goes for any technique, it will only be as effective as the weakest aspect of it. Any one aspect lacking will negatively affect the effectiveness of any technique

Technique

I say it all matters. If a punch isn't accurate it's useless or worse it opens you to a counter attack. If its slow its easy to block. If its weak its ineffective. Etc etc

The point is that there are no unimportant elements in your techniques. They have to all be right or you're wasting your time and counting on luck. I don't know about you but I expect my techniques to work and to do that I train in a way that develops all if the elements.

Proper alignment and correct use of Koshi.

These two aspects alone can give you accuracy and timing beyond the norm.

They will also give you speed and depth of penetration.

Your energy is driven through the opponent.

This is only my opinion.

Edit:

I forgot an equally important thing - BREATH!

I'll keep my answer simple as I can. Speed, power, and the after affect (what the punch sets up for a following series.)

good body mechanics and alignment,