Our family pronounced the name "Dun-a-gun" and for many years we thought it was spelled Dunnigan. On my GGGrandmother's marriage certificate her maiden name was spelled Dunagan in one place and Dunegan in another place on the same document! Research has confirmed the correct spelling of the family name is Dungan. This Dungan family is from Claiborne County, Mississippi.
There is an Anglo-Saxon name Dungen which is from the Germanic word 'dungen' meaning a person who lived on a mound. This goes back to ancient times when many people lived on a mound in the middle of a lake. These people are known to history as the "mound dwellers". Historians think this was a defensive strategy with the lake serving as a moat. This strategy became the model for later forts and castles surrounded by a moat. The meaning of the word "dun" evovled from 'mound' to 'fortified village' to 'castle'. All of these were usually situated on high ground for both drainage and defense.
However, the surname Dungan appears to be mostly Irish being a cognate of the surname Dunnigan, Donigan, Dunagan, etc. You will find the name O' Dungan in Ireland.