1580s, perhaps from Middle Dutch blinken "to glitter," which is of uncertain origin, possibly, with German blinken "to gleam, sparkle, twinkle," from a nasalized form of base found in Old English blican "to shine, glitter" (from PIE root *bhel- (1) "to shine, flash, burn").
Middle English had blynke (c. 1300) in the sense "a brief gleam or spark," perhaps a variant of blench "to move suddenly or sharply; to raise one's eyelids" (c. 1200), perhaps from the rare Old English blencan "deceive."
Originally with a vague and shifting set of meanings, many now obsolete, having to do with motion of the eyes; in earlier use "the notion of 'glancing' predominates; in the latter, that of 'winking'" [OED]. Blink as "to wink" is attested by 1761. Meaning "cast a sudden, fleeting light" is from 1786; that of "shut the eyes momentarily and involuntarily" is from 1858. Related: Blinked; blinking. The last, as a euphemism for a stronger word, is attested by 1914.
1590s, "a glance," of uncertain origin, perhaps from a continental Germanic language; see blink (v.). As is the case with the verb, there is a similar noun in Middle English, from c. 1300, that might represent a native form of the same root. Meaning "action of blinking" is from 1924. From the sense "a flicker, a spark," comes on the blink "nearly extinguished," hence "not functioning" (1901).