The Morgans think it probable that Anglo Norman scribes on seeing the Welsh version Rhys, pronounced and wrote it as Rice. This explains how one might be Dafydd ap Rhys one day and plain old David Price the next. 

A recent visitor thinks it possible his family name changed from Rees to Reece when his forebears landed in America. It is certainly true that most of the early forms were written as Rees but there are examples of the Reece spelling from at least the sixteenth century. 
 
 

J.W