From Intel's SDM, section 2.2.1 specifies that the REX prefix is used to:
But what do the letters REX stand for in the acronym?
This 2002 Hot Chips presentation by AMD expands the acronym on slide 10: "REX (Register Extension)".
Kevin McGrath and Dave Christie, "The AMD x86-64 Architecture: Extending the x86 to 64 bits", Hot Chips 14, August 2002.