The English surname Dampier is an anglicized form of the French Dampierre, which seems to have had its origin in Normandy, France. At least, the overwhelming weight of evidence points to that conclusion. It is a place name, that is, a surname that derives it existence from a location.
In the book, Surnames Are The Fossils of Speech, Samuel Brown indicates that Dampier is of French origin, dom Pierre meaning Lord Peter or Sir Peter. The title DOM, which later became DAM, might come from the Latin Dominus meaning Lord or Master or perhaps even Saint. Pierre, of course, is the French form of Peter. The French place names, Dompierre and Dampierre, are derived to give honor to Saint Peter, the apostle of the Bible. S. Baring-Gould in Family Names and Their Story gives Dampierre as a place name in Seine Inferieuer. There are many others.
According to Charles Wareing Bardsley in his Dictionary of English and Welch Surnames, the English Dampier is a later form of the French Dampierre. He lists Dampier as being derived from the local name, de Dampierre, a place near Dieppe, France, also a place in the department of Orne, both located in lower Normandy, France.