Where does the name of the country Greece came from?

Homer, in the Odyssey uses a name for certain Greek tribes of the period, Graichei, or Graikei. This is an approximate transliteration of the Greek. This ancient name for a sub-group of the Greeks (who actually refer to themselves as Hellenes), was taken up and used by westerners to refer to the whole of the Greeks. Through the centuries, and passing through many languages to reach English, Graichei turned to Greeks, and Greece (probably through some romance language where Graichei would turn to Graecci or some such) their country.