moon (n.)
Old English mona,
from Proto-Germanic *menon-,
from PIE *me(n)ses- "moon, month"
That's a straight line of descent from proto-Indo-European to modern English. No borrowing at any point.
Interestingly, etymonline also points out that the english word "moon" is cognate with the classical Greek word for the moon "mene". This leads nicely into your more interesting, and less wrong, claim:
> the Greek name for the body was Io
This is a little unconventional. I'm not really trained in classical Greek, but I also thought the word was selene.
Doing a simple search of greek dictionary entries through perseus ( http://www.perseus.tufts.edu/hopper/definitionlookup?type=be... ) quickly tells us that there are many, many, many greek words relating to the moon that use a form of selene or mene (aselenos, moonless; selenion, moonlight; hyposelenios, under the moon; triselenos, of three moons; numenia, the new moon; dichomenia, the full moon; skotomene, moonless night; etc. etc. etc.) Further inspection reveals that the term selene never appears in Homer, while mene does, so mene is probably older. (Knowing that mene comes from a PIE root also supports the idea that it's older.)
Io also appears, with a short gloss of "the moon". But the related dictionary entries don't really deal with that sense; they prefer to define Io as the mythological daughter of Inachus. But there is a citation to "Eust.ad D.P.92" in which Io is the "name of the moon at Argos". Wikipedia's entry on Io (the mythological figure) also says, "The ancients connected Io with the Moon", with a citation to "Eustathius of Thessalonica commentary on Dionysius Periegetes, 92", which looks like the same thing as the LSJ cite perseus gives. Wikipedia also cites this to a 10th-century Byzantine encylcopedia which Eustathius is known to have been familiar with, and to a 5th-century Greek philologist. None of those cites go to web resources, and I would be unlikely to get much out of them even if they did, but it does appear that there was a minor strain of referring to the moon as Io in some places at some point.
But selene and mene are clearly better candidates for "the word for the moon", since only they are used to form moon-related words. (Really, there are two words for the moon -- both terms appear as goddesses, but obviously have an ordinary, non-divine sense.) As far as I can see, Io is a mythological allusion, not "the word for the moon". It's even capitalized in the dictionary entry with the moon gloss.
How'd you learn that the Greek name for the moon was supposedly Io? Do you remember?
Finally,
> Luna was the Roman goddess represented by the body.
Sure, that's true, but luna is also just the ordinary Latin word for the moon. You might notice that the same thing happens with both selene and mene. Personification deities are dirt-common in classical literature. Take the goddess who started the Trojan war, Eris. She's referred to (by us) as the goddess of discord. And Eris isn't really a name -- it's just the word for quarreling and strife.
Moon came into Emglish from Danish for 'month'.
Luna was the Roman goddess represented by the body.