1590s, from Latin Elysium, from Greek Ēlysion (pedion) "Elysian field," abode of the blessed after death, where heroes and the virtuous dwell, which is of unknown origin, perhaps from Pre-Greek (a non-IE substrate Mediterranean language). Also used figuratively of a situation of complete happiness.
early 15c. (Chauliac), "remaining traces of a disease," from Old French remanence, remenence, related to remanoir "to stay, dwell, remain" (see remain (v.)). By 1660s in the general sense of "that which remains." The meaning "continuance, permanence" is by 1810 (Coleridge).
"pertaining to the people dwelling on the opposite side of the earth," 1860, from antoeci (plural) "people dwelling on the opposite side of the earth" (1620s), a Latinized form of Greek antoikoi, literally "dwellers opposite," from anti "opposite" (see anti-) + oikein "to dwell" (from PIE root *weik- (1) "clan").