"Mexico" does not mean "bellybutton of the moon." This is a poetic but inaccurate folk etymology.
The name derives from the Nahuatl (Aztec) language:
Mēxihco (pronounced [meːˈʃiʔko]) thus means "Place at the Center of the Moon" or "Place in the Moon's Navel."
The name originally referred to Tenochtitlan, the Aztec capital built on an island in Lake Texcoco (modern Mexico City). The city's central island position in the lake—and its role as the political and cosmological center of the Aztec Empire—inspired the "center/navel" metaphor. The moon association may relate to the lake's reflection or lunar symbolism in Aztec cosmology.
"Navel" (xictli) is sometimes translated as "bellybutton" in popular accounts, and "center of the moon" becomes "bellybutton of the moon" through repeated simplification. While evocative, it's linguistically imprecise.