Guy Steele's 
Jargon File defined brain-damaged as
"[generalization of "Honeywell Brain Damage" (HBD),
a theoretical disease invented to explain certain utter cretinisms in Honeywell Multics] adj. Obviously wrong; cretinous; demented."
The definition has propagated to The New Hacker's Dictionary, Wiktionary, dictionary.com, and other online sources.
Here is Jim Fenton's story on the origin of the term.
I tried sending this story to a number of my Student Information Processing Board (SIPB) contemporaries that I'm still in touch with, but got basically no feedback, so I can't guarantee 100% accuracy (who ever can?), but it should be pretty accurate.
Sometime in the mid 1970s, Honeywell introduced iox_, a new input/output API (although we didn't use the term API) for Multics. It was, to say the least, not universally popular. At one point, Seth Steinberg installed a small cardboard coinbox in the MIT SIPB office, with a sign on it to the effect:
"Help stamp out Honeywell Brain Damage (HBD). Get rid of iox_"
The box collected a very small amount of small change, less than $1.00, mostly in pennies.
A few weeks, a clipping from Tech Talk, the MIT administration newspaper, was added, showing some brain research going on at MIT, with the notation, "Your pennies at work"
Bernie Greenberg adds (02 Jul 2015):
The recounting of the unpopularity of iox_ is accurate.
I remember this donation box; as, for a long while, the only person active at SIPB who was employed by Honeywell (c. 1974, after receiving my Master's), I found it quite offensive. At SIPB prevailed a complex ambivalence about Multics. Forward-looking young people who used Multics daily were knowledgeable about state-of-the-art OS research on ITS and Nicholas Negroponte's "Architecture Machine," and thought less of Multics for not keeping pace, as well as for its (prescient) mature attitude about security, while at the same while admiring the idealism and beauty of Multics' goals and design. They rightly looked down at Honeywell as a conservative, old-style corporation pursuing troglodytic goals such as stability, compatibility, profit, and GCOS, and the phrase "Honeywell Brain Damage" iconified all the ways they perceived the noble Multics hidebound by its foster parent. The SIPB "kids" were ambivalent about Multics, but not about Honeywell.
But for me, who cherished both communities (even more so 40 years later), Multics was a rebel child within Honeywell, and Noel Morris, Tom Van Vleck, André Bensoussan, and Steve Webber were no less rebels and explorers and not one iota less talented than these very bright kids (many of whom became lifelong friends), nor any less interested in pushing the state of the art. We (the Cambridge Honeywell Multicians) loved Multics, not Honeywell. For years I was a liaison between these two communities, justifying each to the other, and the very phrase "Honeywell Brain Damage," insulting not only me but the (slightly) older gurus I idolized, hurt me personally.
Part of the motivation for Multics Emacs was to close this gap (for the SIPB "kids", Emacs was the poster child of what Multics lacked), and so it did.