"flat stone for paving," c. 1600, ultimately from Old Norse flaga "stone slab," from Proto-Germanic *flago- (from extended form of PIE root *plak- (1) "to be flat"). Earlier in English as "piece cut from turf or sod" (mid-15c.), from Old Norse flag "spot where a piece of turf has been cut out," from flaga.
1875, "place a flag on or over," from flag (n.1). Meaning "designate as someone who will not be served more liquor," by 1980s, probably from use of flags to signal trains, etc., to halt, which led to a verb meaning "inform by means of signal flags" (1856, American English). Meaning "to mark so as to be easily found" is from 1934 (originally by means of paper tabs on files). Related: Flagged; flagger; flagging.
Meaning "name and editorial information on a newspaper" is by 1956. U.S. Flag Day (1894) is in reference to the adopting of the Stars and Stripes by the Continental Congress on June 14, 1777.
1540s, "flap about loosely," probably a later variant of Middle English flakken, flacken "to flap, flutter" (late 14c.), which probably is from Old Norse flaka "to flicker, flutter, hang loose," itself perhaps imitative of something flapping lazily in the wind. The sense of "go limp, droop, become languid" is attested by 1610s. Related: Flagged; flagging.