Hugh II of Cyprus was succeeded by Hugh of Lusignan-Antioch, and then claim fell to Walter V of Brienne and his descendants.
He was succeeded by Hugh of Lusignan-Antioch (son of his younger aunt Isabella) as Hugh III of Cyprus, though his heir-general was another first cousin, Hugh of Brienne (c. 12401296), son of Mary of Cyprus, the eldest aunt of the deceased Hugh II. This claim fell to his son Walter V of Brienne and his descendants. 
