"flat end of an arm of an anchor," 1560s, perhaps from fluke (n.3) "flatfish," on resemblance of shape, or from Low German flügel "wing." Transferred meaning "whale's tail" (in plural, flukes) is by 1725, so called from resemblance.
"lucky stroke, chance hit," 1857, also flook, said to be originally a lucky shot at billiards, of uncertain origin. Century Dictionary connects it with fluke (n.1) in reference to the whale's use of flukes to get along rapidly (to go a-fluking or some variant of it, "go very fast," is in Dana, Smyth, and other sailors' books of the era). OED (2nd ed. print) allows only that it is "Possibly of Eng. dialectal origin."
"flatfish," Old English floc "flatfish," related to Old Norse floke "flatfish," flak "disk, floe," from Proto-Germanic *flok-, from PIE root *plak- (1) "to be flat." The parasite worm (1660s) so called from resemblance of shape.