As mentioned in the other answer, this claim was made in 2004 by Bruce Blair, a former Minuteman launch officer. All sources for this story seem to trace back to Blair. He wrote:
Last month I asked Robert McNamara [...] what he believed back in the 1960s was the status of technical locks on the Minuteman intercontinental missiles. [...] McNamara replied [...] that he personally saw to it that these special locks (known to wonks as “Permissive Action Links”) were installed [...] and that he regarded them as essential to strict central control and preventing unauthorized launch.
[...] What I then told McNamara [...] elicited this response: “I am shocked, absolutely shocked and outraged. Who the hell authorized that?” What he had just learned from me was that the locks had been installed, but everyone knew the combination.
[...] During the early to mid-1970s, during my stint as a Minuteman launch officer, [...] [o]ur launch checklist in fact instructed us, the firing crew, to double-check the locking panel in our underground launch bunker to ensure that no digits other than zero had been inadvertently dialed into the panel. [...] And so the “secret unlock code” during the height of the nuclear crises of the Cold War remained constant at OOOOOOOO.
After leaving the Air Force in 1974, I pressed the service [...] to consider a range of terrorist scenarios in which these locks could serve as crucial barriers against the unauthorized seizure of launch control over Minuteman missiles. [...]
The locks were activated in 1977.
According to this January 2014 article at foreignpolicy.com,
A new wave of media coverage [of Blair's column] sparked by online media outlets last year prompted the House Armed Services Committee to ask about the issue, and the military responded by insisting "00000000" was never used.
The actual response from the Air Force is embedded in the article. Here are some excerpts (emphasis theirs):
The MM system has never used Permissive Action Links (PALs). PALs are strictly a nuclear aircraft crew device. Additionally, a code consisting of eight zeroes has never been used to enable or launch a MM ICBM, as claimed by Dr. Bruce Blair.
During the 1970s, 1980s and 1990s [the] second portion of the enable code was dialed into analog six digit thumbwheel switches located on the front of the Launch Enable Control Group (LECG) [...]
A daily check of the visible thumbwheel switches on the front of the LECG ensured that they were set to a standardized setting of “P7.” This provided a standardized “starting point” to help speed the process of flipping the switches in the event an
enable code needed to be entered during time-critical missile launch procedures.
The document includes a photo of the LECG showing the six dials set to P7.
According to the article, Blair disputes the Air Force's claims. It's worth reading the whole article, but here are some excerpts:
[...] Blair [...] told Foreign Policy that [...] the new document [...] leaves out key basics from before 1977. That is when a program known as Rivet Save added in additional security precautions, including new launch codes [...]
"Before this real enable code system was adopted, there was no technical safeguard and both crewmembers were thus required to stay awake throughout the alert period in the underground capsule," Blair said. "Thus the document errs and misleads when it says that the 00000000 enable code system was never used."
Attempts to solicit comment from the Air Force were unsuccessful. However, Lance Lord, a retired four-star general and former nuclear launch officer, said he does not recall any codes including all zeroes ever being used. Like Blair, he recalled that both crew members in a launch control center were required to stay awake prior to Rivet Save being put in place in the late 1970s.
Below this point is only my own speculation. I'm inclined to disbelieve Blair for a few reasons:
I can't find evidence that any other former launch officers agree with him, and the article quotes one who disagrees.
The fact that the Minuteman system never used PALs makes me question his knowledge of the system. (I have no independent knowledge of this, but I'm assuming the Air Force wouldn't lie to Congress about something so easily checked.) The eight zeros versus six P7s discrepancy is also odd, but it could be related to the 1977 changes.
"The locks were activated in 1977" seems to refer to the deployment of a new security system, not enabling a neglected part of an old one.
Presuming Blair never had to enter a real enable code, it seems plausible that he might have misinterpreted the reason for setting the dials to a constant value, when it was actually just to make it easier to enter the real code, as the Air Force claimed.
McNamara may have reacted as described, but of course there's no guarantee that what Blair told him was true.