Though the origin of the name “Europe” is not known for certain, the name is very old. A Greek hymn to Apollo from the sixth century B.C. mentions the name, which originally applied only to part of the Balkan Peninsula. ‘Europe probably derives from a word meaning "mainland," though it may be related to the Assyrian “ereb” –“darkness, west," and therefore "land of the setting sun."
“Asia” probably got its name from the Assyrian “asu” --"sunrise, east." “asu”originally referred only to the east coast of the Aegean Sea but gradually came to include the whole continent. (The Book of Answers, by Barbara Berliner)