The first speech uttered from the surface of another celestial body turned out to be an absurdity. The task fell to Armstrong as he descended the stairs of the lunar module: ‘That’s one small step for man, one giant leap for mankind.’ But ‘man’ and ‘mankind’ amounted to much the same thing; if there was to be any point in the First Sentence, it would derive from an implied contrast between what a particular individual did and its significance for the whole of humanity. After a few weeks, Nasa could no longer withstand repeated observations that the First Sentence was vacuous. Armstrong said that he was ‘misquoted’ in the official transcript and an official spokesman announced that ‘static’ obscured a missing ‘a’: ‘That’s one small step for a man, one giant leap for mankind.’ ‘I rehearsed it that way,’ Armstrong later said. ‘I meant it that way. And I’m sure I said it that way.’ The claim, however, smacks literally of l’esprit de l’escalier, and you can judge its accuracy for yourself by listening to the recording at www.hq.nasa.gov/office/pao/History/alsj/a11/a11.step.html. There is no evident ‘static’ and it’s clear enough that Armstrong said what everybody heard him say at the time. In the event, he eventually gave up the pretence: ‘Damn, I really did it. I blew the first words on the Moon, didn’t I?’

From a review of Moondust: In Search of the Men Who Fell to Earth by Andrew Smith in the London Review of Books. Lots of great anecdotes (in the review, and, I'm sure, in the book itself).

Behind the scenes was enough drama for an HBO reality show:

Despite the vast attention paid to the astronauts’ psychological profiles and their ability to work in teams, the Apollo 11 crew verged on the dysfunctional. While Armstrong and Aldrin didn’t quite match Stoppard’s Scott and Oates, there was a fierce behind-the-scenes battle between them to be first to set foot on the Moon. Early plans were for Aldrin, as module pilot, to step out first, but one version reported by Smith has it that Armstrong, as mission commander, lobbied more vigorously than Aldrin, and Nasa backed him up because he would be ‘better equipped to handle the clamour when he got back’ and, more mundanely, because his seat in the lunar module was closer to the door. Aldrin paid Armstrong back by taking no photographs of him on the Moon: the only manually taken lunar image of the First Man on the Moon is in one of many pictures Armstrong snapped of Aldrin, showing himself reflected in the visor of Aldrin’s spacesuit. Asked about this omission later, Aldrin lamely replied: ‘My fault, perhaps, but we had never simulated this in training.’ Later, Aldrin put it about that Armstrong’s First Sentence might have been a bureaucratic concoction.