(JUN-jee) pronunciation can be heard here: http://www.forvo.com/search/Gyöngyi/
The name comes directly from the Hungarian word for pearl. Its designated name-day in Hungary is October 23. However, the pearl is the birthstone of June. Another name-day in Hungary is May 14. There is also an older form of the name Gyöngyvér (JUN-jee-VARE), which is an old poetic word for pearl. According to Hungarian legend, Gyöngyvér was the wife of Buda (the brother of Attila the Hun).