The name of the horehound (hoarhound) herb is really contained in the second element, which, though considerably altered through the centuries, seems to have been earliest in use.
That is, in England of about a thousand years ago, the name of the plant seems to have been hune, and, to distinguish the one covered with white cottony hairs and small white flowers from others similar, but less attractive, it was described as the hare hune, from har, “hoary, white.”
Both elements underwent change, hune becoming houne in the fourteenth century, hounde in the fifteenth, and hound in the sixteenth, and there’s no association whatsoever between the plant and any member of the canine family.