Answer:
I have seen it spelled both ways and at first I thought "whet" was the olde english spelling. I hope you find this explanation as sensible as I did.
To whet is to sharpen / hone. So harken back to the days before modern machinery. It's harvest time and the gentlemen farmers would gather in each other's fields to cut grain with their long scythes, also called whistles as they whistled as the men cut through the crops. Obviously, as the men stopped to sharpen their scythes or whet their whistles, the whistling would stop. This would alert the women folk back at the farm house, that the men had stopped and it was a good time for the ladies to bring them water. So the whet your whistle is to sharpen a scythe. To wet your whistle is to satisfy the need for a drink. Ever tried whistling with a dry mouth?