Hamilton’s most famous contribution to the Apollo program came during the pivotal Apollo 11 mission to land on the moon in July of 1969. Moments before the Eagle module was set to land on the lunar surface, the onboard computer raised the alarm. It had detected a 1202 error: The computer was overloaded due to a fault in a hardware switch, and it was possible the system would not be able to handle the complex landing procedure. 

But Hamilton and her team had engineered priority-driven software, able to shut down unnecessary background tasks in order to prioritize mission-critical tasks. Houston trusted Hamilton’s software and allowed the mission to proceed, and two men walked on the moon for the first time.

This is clearly depicted in the excellent film 'First Man' (2018) during the tense final landing when Neil Armstrong is visibly annoyed at the alarm but Houston, after a short delay, allow it to be switched off.