Paul by Adrian/ Hadrian searching for the root.... Read on
Good question. First of all: I found that the Adriatic Sea got its name from a different town with the same name as the one, Hadrianus derives from. "According to ancient tradition, the byname of Emperor Publius Aelius Hadrianus goes back to the place of origin of his parents, Hadria, Atri today, Province of Teramo (not Adria south of Venice)." (Rosa and Volker Kohlheim, Das Große Vornamenbuch, Mannheim 2003)
Very little did I find about the origin of the name of either town. Der Kleine Pauly, Lexikon der Antike, München 1979 has (obviously referring to Hadria in central Italy: "Latinic settlement (colony), 6 m from the shore, founded 298 BC, with an insignificant port by the same name. On coins there is the name HAT, which goes well with Stephanos Byzantinos' ATRIA, he claims it was founded by a man called Diomedes whose real name was AITHRIA." In Greek, AITRIA means "bright sky." Der Kleine Pauly also confirms that the Adriatic Sea has its name from ATRIA near Venice, but it does not give a meaning.