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.
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.
Best Answer
Garrison Keillor doesn't know, and online sources are all rather contradictory (the game is variously said to have originated in Germany, Ireland, Sweden, and the U.S.) and dubious. So the best answer may well be that origin of the Minnesota name is lost to history, and any answers will be speculative.
I noticed, however, that of all the variants played in different countries— le facteur ("the postman"), plumpsack (lit. "the plump sack"), 수건 돌리기 (loosely, "towel whirl")— only the Swedish name is similar to the English: Anka Anka Gås (i.e. duck duck goose). Digging around a bit more, I did find a site which referred to the Swedish game by a different name: Anka Anka Grå Anka, which happens to translate to duck duck grey duck. An alternative, Anka, Anka, Gråttanka, is attested to on Reddit.
Now, this name is vanishingly rare on the Internet compared to Anka Anka Gås (4 vs 27,000 Google.se results). Nevertheless, Minnesota has the largest concentration of ethnic Swedes in the U.S.— its most common surnames to this day are Scandinavian, and I could certainly envision a scenario where a regional or dialectic name of the game was transplanted by immigrants to became the prevalent usage in the new country even as it faded in the old. The Swedish community was fairly isolated until the turn of the 20th century but has been economically quite successful since.