Thanks to Delta for pointing out the location of Elfin Chain in the DMG (p. 27):

Chain, Elfin,  is a finely wrought suit of chain which  is of  thinner  links but stronger metal. It is obtainable only from elvenkind who do not sell it.

It is mentioned once more on page 28, in a section about magical armor:

When magic armor  is worn, assume  that  its  properties allow movement of  the  next higher  base  rate and  that weight  is  cut  by 50%.  There  is no magical elfin chain mail.

Besides a comment about it being immune to the Heat Metal spell and a row in the encumbrance chart, those are the only references to elfin chain in the DMG.  As such, I think it’s pretty clear that it was not intended to be handed out to players.  It’s explicitly not purchasable per the description, nor does it show up in the treasure charts.  However, as we can see in A1, this did not deter players from obtaining it and DMs from handing it out.  As such, we see it show up in the treasure charts of Unearthed Arcana.  What’s more, we see magical versions of to +5 enchantment (Unearthed Arcana, p. 88), which directly contradicts the second quote above from the DMG.

Delta claims that this ultimately started as an attempt to explain the unusual movement rate of elves in Chain Mail.  It’s a reasonable explanation, and I find it thus fascinating how strongly the idea took hold in the D&D community.  Clearly this item is the grand-father of mythril, a concept now thoroughly enmeshed in the fantasy genre.  Sure, the true source here is probably the armor Tolkien gave to Bilbo, but I suspect that the fantasy-consuming culture at large is responsible for its increased significance.  I’m tempted to go back and re-read the parts focusing on this armor in The Hobbit and Lord of the Rings, to see if as much significance is given to it as in the movies.  I suspect not.