"to place (papers) in consecutive order for future reference," mid-15c., from Middle French filer "string documents on a wire for preservation or reference," from fil "thread, string" (12c.), from Latin filum "a thread, string," from PIE *gwhis-lom (cf. Armenian jil "sinew, string, line," Lithuanian gysla "vein, sinew," Old Church Slavonic zila "vein"), from root *gwhi- "thread, tendon." The notion is of documents hung up on a line.
Methods have become more sophisticated, but the word has stuck. Related: Filed; filing.