That region is Cilicia, which was also called Lesser Armenia in medieval era, since numerous Armenians fled there when Armenia proper fell under Seljuk Turkish rule. When the First Crusade arrived from the west, the crusaders destabilized Anatolia enough so that local Armenian nobles managed to defeat the Byzantine and Turkish forts in Cilicia, with Crusader help, and in the aftermath the Armenians set up their own principality, and later kingdom, which was to last until 1375.

Some historical sources have named the kingdom or principality as the Armenian Kingdom of Cilicia or variants thereof, to distinguish it from the Bagratid Armenian kingdom that existed in Armenia proper from 885 to 1045 or the Armenian kingdom of antiquity.

Writing from work, so have no sources to come with more detailed information for the time being. Also, it is not a topic I am very familiar with (used Wikipedia to double-check dates), so hopefully someone more knowledgeable in Armenian matters can give a more detailed response.