Dempsey is an Anglicized Irish surname meaning "proud," from "son of Ó Diomasaigh. The family with this name originated in the Kingdom of Uí Failghe. On occassion, this name was simply given as "Proudman" in English. This is not a unisex name.
The title Viscount Clanmalier was created for Terrance O'Dempsey by King James I, and after him there were only two others to hold the title before it was no longer used - Lewis O'Dempsey and Maximilian O'Dempsey. The O'Dempsey family was strong and powerful until the 17th century, or, until the Williamite wars. Several prominent people have bared this surname, including feminist activist Rosemary Dempsey, but two namesakes spring to mind immediately - the actor Patrick Dempsey, proud father of Sullivan and Darby, and Jack Dempsey, the 20th century boxing champ. There are dozens more, however, listed all over the web. Only a small handful feature Dempsey as a given name. In my opinion, Dempsey would make a fine first name for someone from this family tree, especially if it was the mother's maiden name.
In 2011 Dempsey was only given to 67 boys. It first appeared on the charts in 1880 at #579, which would suggest that if these charts had been up years before, it would show Dempsey gaining or continuing popularity until 1880. However, it was on and off the charts for the entire time it ranked, up until 1949, and it hasn't been back since.