Old English læt "occurring after the customary or expected time," originally "slow, sluggish," from Proto-Germanic *lata- (cf. Old Norse latr "sluggish, lazy," Middle Dutch, Old Saxon lat, German laß "idle, weary," Gothic lats "weary, sluggish, lazy," latjan "to hinder"), from PIE *led- "slow, weary" (cf. Latin lassus "faint, weary, languid, exhausted," Greek ledein "to be weary"), from root *le- "to let go, slacken" (see let (v.)).
The sense of "deceased" (as in the late Mrs. Smith) is from late 15c., from an adverbial sense of "recently." Of women's menstrual periods, attested colloquially from 1962. Related: Lateness. As an adverb, from Old English late.