Emacs originally was an acronym for Editor MACroS. RMS says he “picked the name Emacs because E was not in use as an abbreviation on ITS at the time.” The first Emacs was a set of macros written in 1976 at MIT by RMS for the editor TECO (Text Editor and COrrector, originally Tape Editor and COrrector) under ITS (the Incompatible Timesharing System) on a PDP-10. RMS had already extended TECO with a “real-time” full-screen mode with reprogrammable keys. Emacs was started by Guy Steele as a project to unify the many divergent TECO command sets and key bindings at MIT, and completed by RMS.
Many people have said that TECO code looks a lot like line noise; you can read more at news:alt.lang.teco. Someone has written a TECO implementation in Emacs Lisp (to find it, see Packages that do not come with Emacs); it would be an interesting project to run the original TECO Emacs inside of Emacs.
For some not-so-serious alternative reasons for Emacs to have that name, check out the file etc/JOKES (see File-name conventions).