English (also present in Ireland): from Middle English peni, peny ‘penny’, applied as a nickname, possibly for a person of some substance or for a tenant who paid a rent of one penny. This was the common Germanic unit of value when money was still an unusual phenomenon. It was the only unit of coinage in England until the early 14th century, when the groat and the gold noble were introduced, and was a silver coin of considerable value. There is some evidence that the word was used in Old English times as a byname.
According to a grandfather, Rommie Penny, the family was to have originated in Scotland. And according to "Scottish Kith and Kin" the Penny name is a sept of the Morray clan. I have also heard that the name has ties to a line of English nobility.
So, there you have it!