That's an interesting question. Wikipedia doesn't say. [1] I did a brief Google Books search and found use of "register file" dating back to 1966. I assume it's related to filing-cabinet file, not file system file.
Etymology of the word "file" is interesting [2]. It comes from Middle French filer meaning 'string documents on a wire for preservation or reference'. That's surprisingly close to what a computer's registers do, right down to using wires :-)
Etymology of the word "file" is interesting [2]. It comes from Middle French filer meaning 'string documents on a wire for preservation or reference'. That's surprisingly close to what a computer's registers do, right down to using wires :-)
[1] http://en.wikipedia.org/wiki/Register_file
[2] http://www.etymonline.com/index.php?term=file