The French spelling of this male name is
Rene. Click on the linked name:
Rene for its derivation.
According to Hanks & Hodges's
A Dictionary of Surnames, Reyne is a French surname, pronounced "rain", which goes back to the medieval feminine given name
Reine, meaning "queen".
So it seems that if your parents or someone further back got confused between
Rene and Reyne. They are both French names, but they do not have the same pronunciation.