1570s, "one who sails along coasts," especially one who trades from port to port in the same country, agent noun from coast (v.) in its sense "to go around the sides or border" of something. Applied to vessels for such sailing from 1680s.
The meaning "tabletop drink mat" to protect a wooden table surface from condensation, etc., was in use by 1913, extended from bottle-coaster "low, round tray used for a decanter" (1874); it was formerly on wheels and so called probably because it "coasted" around the perimeter of the table to each guest in turn after dinner.
