Latin gaudium travelled through French joie into English as joy, and gave the Romance languages a family of names. It sits alongside laetitia, gladness, which produced Letizia and Letitia.
Hebrew simḥah is used for both men and women and denotes joy of a specifically celebratory kind — the word for a happy occasion as well as for the feeling. Arabic faraḥ and surūr divide the field between delight and inner contentment.
The West African material on this page is worth reading closely. Yoruba ayọ̀ gives a large family of names, and many of them are full sentences: statements that joy has arrived, or that the child is the joy, or that joy has replaced something else. That is a different grammar of naming from the European habit of using an abstract noun on its own, and it means a Yoruba joy-name usually tells you something about the family's circumstances at the time of the birth.
Sanskrit ānanda — bliss — carries a philosophical sense of joy as a property of ultimate reality rather than a passing mood, and names built on it are religious in origin.