Where did the word 'Greece' come from?

The Greeks I've met say something that to my ears sounds like 'Elatha'.

Solution 1:

What you are hearing is most likely the correct Greek pronunciation of Elláda (Ελλάδα). This is the modern Greek word for the name of their country, ultimately derived from the Ancient Greek Hellás (Ἑλλάς).

The English name for the country, "Greece", derives from the Latin name "Graecia". Wiktionary gives a fairly full etymology:

From Latin Graecia < Ancient Greek Γραικός (Graikos), a character in Greek mythology, the son of Thessalos, the king of Fthia, from whom Ἑλλάς (Hellas, “Greece”) and Ἕλληνες (Hellenes, “the Greeks”) got their names.

Although this entry explains the etymology of the name "Greece", it is admittedly slightly confusing about the etymology of "Hellas". This page gives a hypothetical etymology:

Etymology: From Ancient Greek (Hellas "Greece"), from prefix - (el-ελ "sun, bright, shiny", (elios, "sun")) + (las-λας "rock, stone"). : "The land of the sun and the rock".

I would not however want to comment on the veracity of this source. All that is known for sure is that Hellas originally referred to a small area within Ancient Greece and only later came to refer to all Greece. This Yahoo answer gives some handy details.

Solution 2:

Greece, as the name of a country, was more used by outsiders than Greeks, at least in the Hellenistic period. Most of them would think of themselves as Athenians or Corinthians, and Hellas was not the place where they lived, but the area where people spoke Greek as opposed to barbarian languages; similar to mediaeval Christendom. (The Greek colonies in Italy were referred to as 'Megale Hellas' or Greater Greece). It was only when Greece became part of the Roman Empire that a name was needed for the geographical area, and unsurprisingly the Roman word Graecia was used. It appears that Germany had a similar history: Why does Germany's English name differ from its German name?