Sure we did - that name is "The Moon". It's even the official recognized name by the International Astronomical Union, which is the body responsible for such things.
Note that the definitive article is important. It's not just a moon, it's The Moon.
Cruithne be damned.
Technically it's the capitalization that makes it the proper name - "Moon" vs "moon". To be pedantic, the IAU's official name is just "Moon", not "The Moon". You can read more here:
1976: The names of the then major planets (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto) and the Earth's satellite (Moon) appear in IAU Resolution No. 10, which was approved by the XVIth General Assembly of the IAU in Grenoble, France in 1976.
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So the IAU does recognize official names for the major planets (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune) and Earth's satellite (Moon).