The Greeks I've met say something that to my ears sounds like 'Elatha'.
Learn English – Where did the word ‘Greece’ come from
etymology
Related Solutions
Edit: found the citation from 1672, from Andrew Marvell’s The Rehearsal Transpros'd:
Two or three brawny Fellows in a Corner, with meer Ink and Elbow-grease, do more Harm than an Hundred systematical Divines with their sweaty Preaching.
It's also defined in B.E.'s A New Dictionary of the Terms Ancient and Modern of the Canting Crew, in its several tribes, of gypsies, beggers, thieves, cheats, &c. with an addition of some proverbs, phrases, figurative speeches, &c., c.1698:
Elbow-greaſe, a deriſory term for Sweat. It will coſt nothing but a little Elbow-grease ; in a jeer to one that is lazy, and thinks much of his Labour.
I found no earlier mentions than senderle, but here are some useful references. These are the earliest references I could find, and helpfully, they are also dictionary definitions.
The Online Etymology Dictionary says
Phrase elbow grease "hard rubbing" is attested from 1670s, from jocular sense of "the best substance for polishing furniture."
There's a similarly colourful definition in Francis Grose's 1785 A Classical Dictionary of the Vulgar Tongue:
ELBOW GREASE, labour, elbow grease will make an oak table shine.
(The rest of this dictionary is interesting too!)
Also, very pertinent to the question, here's The Royal Dictionary, French and English, and English and French by Abel Boyer in 1729:
Elbow-grease, (or Pains) Rude travail.
Rude travail is French for rough work. There's no entry for "l'huile de coude" in the French side.
And in John S. Farmer and W.E. Henley's 1905 A Dictionary of Slang and Colloquial English:
Elbow-grease. Energetic and continuous manual labour : e.g. Elbow-grease is the best furniture oil : Fr., huile de bras or de poignet ; du foulage (1779).
French huile de bras or de poignet is oil of the arm, wrist which is quite close. I think du foulage is fulling, the manual scouring and milling of cloth.
The earliest French reference I could "l'huile de coude" helpfully explains the term. In Jean Humbert's 1852 Nouveau Glossaire Genevois: Volume 1 (New Geneva Glossary):
Dans le langage badin des domestiques et des maîtresses, l'huile de coude, c'est le frottage, c'est-à-dire : Le travail de la servante qui frotte. Ces meubles, Madame, ne veulent pas devenir brillants. — C'est que, ma mie, tu y as sans doute économisé l'huile de coude; c'est-à-dire : Tu as trop ménagé ton bras et tes forces.
A rough translation:
In the playful language of servants and masters, elbow grease is rubbing, i.e. the work of the maid who scrubs. This furniture, Madam, does not want to shine. - My dear, that is because you have undoubtedly skimped on the elbow grease. In other words, you have conserved both your arm and your strength.
These references also suggest that "l'huile de coude" is an anglicisme.
Urban dictionary has some surprisingly good entries on the topic if you ignore the humour surrounding it:
This phrase draws an analogy to the poker ante (two bits) and gains your entry into the conversation.
The trick is recognising the (I assume) older bits instead of cents.
Also, two-bit still lives on in common usage, meaning "insignificant":
That's my insignificant contribution.
Best Answer
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:
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:
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.