Old English rima "edge, border, verge, coast," as in særima "seashore," literally "rim of the sea," and dægrima "dawn," literally "rim of the day." Related to Old Norse rime, rimi "a raised strip of land, ridge," Old Frisian rim "edge." "There are app. no parallel forms in the other Teutonic languages" [OED]. but with no other known cognates.
As "the circular part farthest from the axis of a wheel," c. 1400. The general sense of "border or edge of anything," typically a circular border raised above the enclosed surface, is by c. 1600. The snare drummer's rim shot (striking the rim and the head at once) is recorded from 1934. In political geography, rimland for "peripheral region of political or strategic significance" is by 1944.