"First Man" by James R. Hansen, explains,
In response to the reporter’s question about choosing his first words from on the lunar surface, Armstrong simply answered, "No, I haven’t." As hard as it may be to believe, that was the plain truth. "The most important part of the flight in my mind was the landing," Armstrong explains today. "I thought that if there was any statement to have any importance, it would be whatever occurred right after landing, when the engine stopped. I had given some thought to what we would call the landing site. I had also thought about what I would say right at the landing; I thought it was the one that history might note. But not even that was something that I had given a great deal of thought to, because, statistics aside, my gut feeling was that, whereas we had a ninety percent chance of returning safely to Earth, our chances were only even money of actually making the landing."
In fact, Neil had already chosen Tranquility Base as the name of the spot on the Sea of Tranquility where he and Aldrin would land; privately, he had told Charlie Duke about the name, since Duke would serve as CapCom during the landing and Neil did not want Charlie to be caught unawares when Neil used the phrase immediately upon touchdown. "In the absence of official names for the various locations and landmarks on the lunar surface," he told the press, "we have chosen to use some unofficial names for our recognition purposes and for our training purposes, and we’ll continue to do that." No one else in NASA besides Charlie Duke knew about Tranquility Base until Eagle landed.
As they landed in the moon feature "Sea of Tranquillity (Mare Tranquillitatis)" Armstrongs naming seems more than apt.