The name "monoid" was first used in mathematics by Arthur Cayley [*] for a surface of order $n$ which has a multiple point of order $n-1$.

In the context of semigroups the name is due to Bourbaki [source, page 30]

It is also worth commenting on the related term monoid, meaning an
associative magma with identity. This term is a little more recent
than semigroup, and seems to originate with Bourbaki [**]. Before
this, Birkhoff (1934) was using the term groupoid for an associative
magma with identity.

Dov Tamari [source, page 1] argues that Bourbaki "probably intended to abolish the term semigroup for reasons of linguistic taste."

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[*] A. Cayley, Second and Third Memoirs on Skew Surfaces, Otherwise Scrolls, Phil. Trans. (1863 and 1869).
[**] N. Bourbaki, Éléments de Mathématique, Algèbre, Hermann, Paris (1943): Chapter I, §2.
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