
Per the Lore Book, the short term is just "common" which refers to "Hyuran Common Tongue". During the Hyuran migrations the language spread all over Eorzea and since the Hyur were easily one of the most populace race, the tongue caught on and eventually became the commonly spoken language in the land, leading to the death knell of other tongues in Eorzea such as the ancient Elezen tongue. As I'm at work I don't have a scan of the page with this on me but I believe my memory is correct.