"calm, quiet, peaceful," 1540s, in halcyon dayes (translating Latin alcyonei dies, Greek alkyonides hemerai), 14 days of calm weather at the winter solstice, when a mythical bird (also identified with the kingfisher) was said to breed in a nest floating on calm seas. The name of this fabulous bird is attested in Middle English as alcioun (late 14c.).
The name is from Latin halcyon, alcyon, from Greek halkyon, variant (perhaps a misspelling) of alkyon "kingfisher," a word of unknown origin. The explanation that this is from hals "sea; salt" (see halo-) + kyon "conceiving," present participle of kyein "to conceive," literally "to swell" (see cumulus) probably is ancient folk-etymology to explain a loan-word from a non-Indo-European language. Identified in mythology with Halcyone, daughter of Aeolus, who when widowed threw herself into the sea and became a kingfisher.