I believe this comes a long way, from the "pre-http" era. Either Usenet, or even BBSs. Maybe as early as around 1987?...
I remember that there was tons of ascii-art circulating in the early days of Usenet. And IIRC in one of them it started to feature a cow, then some other posts featured more cows, then a post was entirely dedicated to several cows ascii-arts. I believe this easter-egg comes from someone reading those at that time...
I did a few seaches and found out a geocities page talking about it. That page states (excerpt:)
Besides digrams and charts, probably the earliest ASCII art from the Internet
are the "Spy at the Wall" collection and the "Silly Cows" collection.
David Bader, an ASCII art enthusiast and editor of the 'Cows", recently sent
me the COMPLETE, UNCUT, ORIGINAL, and OFFICIAL Silly Cow collection!
These cows can be seen all over the Internet and are truly considered to be
"classic" ASCII art..
with "Silly cows" linking to : http://www.geocities.com/spunk1111/cows.htm (also available on the Internet Wayback Machine at : https://web.archive.org/web/20131225210911/http://www.geocities.com/spunk1111/cows.htm , or go rather to https://web.archive.org/web/*/http://www.geocities.com/spunk1111/cows.htm and click in the agenda below on the day of the Snapshot you want to see...)
Of course a true reference lies in usenet archives, but I don't have much time yet to do proper research (I may update this post in the near future)
At some point there was even a
alt.cows.moo.moo.moo newsgroup created (probably quite a bit after cows started to invade ascii arts? But maybe before, I lack time to research properly) (see for example : http://www.418-teapot.com/topics/usenet/ )
To prove how popular it was in Usenet, the first question mentionned on the Internet Oracle wikipedia page: http://en.wikipedia.org/wiki/Internet_Oracle is about cow(s).