Between starting the descent burn three seconds late, and having to avoid landing in a boulder field, the Apollo 11 descent module almost completely missed the target landing ellipse.
During Apollo 11's stay on the surface, Michael Collins was given seven different locations to look for the Eagle, all of them wrong. Additionally, an attempt to ping the Lunar Ranging Retro-Reflector shortly after deployment was unsuccessful.
When did NASA figure out exactly where the Eagle had landed?
They used the rendezvous radar on the LM to pinpoint the CSM's relative direction, and worked backwards from that info and the CSM's location in order to pinpoint the LM.
Here's the commentary from the Apollo 11 Lunar Surface Journal at 121:00:34:
[In a 2010 book, From the Trench of Mission Control to the Craters of the Moon, H. David Reed, who was Flight Dynamics Officer (FIDO) for the Apollo 11 LM liftoff, details the story behind this unscheduled request for a P22 one orbit before liftoff. Briefly, he and his support team needed an accurate LM position so they could pick a liftoff time that would minimize propellant usage. The various estimates available at that time were scattered over a considerable area. They needed something better. After extensive discussions within the team, Reed choose a method suggested by Pete Williams, the COMPUTER DYNAMICS officer: they would track the CSM with the LM's rendezvous radar and, then, using a separate, accurate determination of the CSM's orbital track over the landing site (as discussed at 121:07:37), work backwards to find the LM.]
The LM P22 operation, tracking the CSM with the rendezvous radar, was started at 122:15. Seven minutes later mission control reported they had the data they needed:
122:22:04 Evans: Tranquility, affirmative, and we're saving it. (Pause) We've got four (data) points so far, and it's looking good.
Half an hour later, mission control gives Eagle the ascent "pad", i.e. the data needed by the computer to fly the ascent, and this is not revised prior to the launch, so it presumably incorporates the location estimate taken from the radar observation:
122:50:39 Evans: Tranquility Base, Houston. Roger. (Long Pause) Columbia, Houston. Over. (Long Pause) Tranquility, Houston. I have your LM ascent and CSI data Pads when you are ready to copy.
122:51:33 Aldrin: We're ready to copy.
122:51:36 Evans: Roger. LM ascent Pad: TIG, 124:22:00:00; Noun 76, 55349, 00322, plus 0017; DEDA 47, plus 37104, minus 70470, plus 58604, plus 56936. Your LM weight 10837. Your T14 (is) 126, plus 20, plus 12. Over. [If, for some reason, they miss the planned launch opportunity, T14 would be the launch opportunity for the next CSM pass over the landing site. ]
122:53:18 Armstrong: What's...Say again the crossrange in Noun 76.
122:53:26 Evans: Roger. Your crossrange for Noun 76...By the way, we may update this later, but now it is plus 0017. Over.
122:53:47 Armstrong: Roger. Readback follows. TIG 124:22:0000, 55349, 00322, plus 0017, plus 37104, minus 70470, plus 58604, plus 56936. LM weight 10837. T14, 126:20:12. Go.
The pad data is not updated after that. An hour later, mission control gives Collins their best guess at the location:
123:55:23 Evans: Columbia, Houston. Roger. Loud and clear. And if you would like to take it down, we have the latest position of Tranquility Base. Over.
123:55:33 Collins: Go ahead.
123:55:36 Evans: Roger. It's just west of West Crater, Juliett 0.5, 7.7. Over.
And this location turns out to be within about 200 meters of the actual position of the LM. The LM ascent stage launched about a half hour later at 124:22.
The actual location seems to have determined sometime after the mission. The November 1969 Apollo 11 Mission Report includes (poorly reproduced) photos with the landing site marked; it should have been possible to work out the location exactly from Lunar Orbiter photos, descent footage, and photos taken on the surface.
