The name Koizis/Koyzis has nothing to do with french origin.
There is a Koizis in Eptakomi andAghios Theodoros of Karpass, a Koizos in Koma tou Yialou and a Koizaris in Trikomo.
The word originates from turkish "kuyu"= well so the name is occupational i.e. the bearer was empoyed as a well-digger a popular job in Cyprus until the late 195o's. In fact the Koizaris in Trikomo was also known as "lakkotripis" which literally means a well-digger and indeed he was.