"the sediment of liquors, foreign matter that subsides to the bottom of a vessel containing liquors," c. 1300 (implied in surname Dryngedregges), from Old Norse dregg "sediment," from Proto-Germanic *drag- (source also of Old High German trestir, German Trester "grape-skins, husks"), from PIE *dher- (1) "to make muddy."
Replaced Old English cognate dræst, dærst "dregs, lees." Figurative use for "useless residue, the most worthless part of anything" is from 1530s.