Nameberry says that "Barron" means "noble fighter," but it can also be roughly translated as "Do you even know who my father is?!"
Sidenote: during my stint at boarding school, Barron's younger brother with an equally pretentious name, Conrad, had a private Facebook group called "Conrad Hilton's Grey Goosers" and a few of my classmates were in it. Dude was only 12 or 13 when he created it. Kind of interesting to see how he ended up, huh?