word-forming element, used from mid-19c. (first in Eocene) in compound words formed by earth-scientists, and meaning "characterized by the earliest appearance of," from Greek ēōs "dawn, morning, daybreak," also the name of the goddess of the morning, from PIE root *aus- (1) "to shine," especially of the dawn. Piltdown Man, before exposed as a fraud, was known as Eoanthropus.
"first appearance of light in the morning," late 13c., verbal noun from dawn (v.). It superseded Middle English dauing, dawing, dayinge, from Old English dagung.
"southern, of or pertaining to the south," 1540s, from Latin australis, from auster "south wind; south," from Proto-Italic *aus-tero- (adj.) "towards the dawn," from PIE *heus-tero- (source also of Sanskrit usra- "red; matutinal," usar-budh- "waking at dawn;" Greek aurion "tomorrow;" Lithuanian aušra "dawn;" Old Church Slavonic jutro "dawn, morning; tomorrow;" Old High German ostara "Easter"), from PIE root *aus- (1) "to shine," especially of the dawn.
The Latin sense shift in auster, if it is indeed the same word other Indo-European languages use for "east," for which Latin uses oriens (see Orient (n.)), perhaps is based on a false assumption about the orientation of the Italian peninsula, "with shift through 'southeast' explained by the diagonal position of the axis of Italy" [Buck]; see Walde, Alois, "Lateinisches etymologisches Wörterbuch," 3rd. ed., vol. I, p.87; Ernout, Alfred, and Meillet, Alfred, "Dictionnaire étymologique de la langue latine," 2nd. ed., p.94.
Or perhaps the connection is more ancient, and from PIE root *aus- "to shine," source of aurora, which also produces words for "burning," with reference to the "hot" south wind that blows into Italy. Thus auster "(hot) south wind," metaphorically extended to "south."