"thing that rubs" (a brush, cloth, etc.), 1530s, agent noun from rub (v.). The meaning "elastic substance from tropical plants" (short for India rubber) first recorded 1788, introduced to Europe 1744 by Charles Marie de la Condamine, so called because it originally was used as an eraser.
Meaning "overshoe made of rubber" is 1842, American English; slang sense of "condom" is by 1930s. Sense of "deciding match" in a game or contest is 1590s, of unknown signification, and perhaps an entirely separate word. Rubber stamp (n.) is from 1881; figurative sense of "institution whose power is formal but not real" is from 1919; the verb in this sense is from 1934. Rubber cement is attested from 1856 (from 1823 as India-rubber cement). Rubber check (one that "bounces") is from 1927.