History of the Frobenius Endomorphism?

Since you reach back to Euler, who proved Fermat's little theorem in the form $a^p \equiv a \bmod p$ by using induction on $a$ and the binomial theorem, I think your "Frobenius endomorphism" is the $p$th-power map in characteristic $p$ (or $p^k$-th power map if the base field has order $p^k$).

Frobenius has his name associated to this rather elementary operation, used long before him, because he proved the existence of lifts of the $p$-th power map to finite Galois groups over $\mathbf Q$. Those automorphisms, which Francois mentions in his answer as Frobenius substitutions, are more intricate than the $p$-th power map in characteristic $p$ and it is not surprising that the person who first published a paper about them got them named after him. (Dedekind wrote to Frobenius that he had gotten the existence of such lifts to Galois groups over $\mathbf Q$ earlier, but this was just a private communication.) Once the name Frobenius substitution was used, it is not surprising that the map in characteristic $p$ got named after Frobenius too. However, I don't know who first used his name for the characteristic $p$ operation.

This is traced in Hasse (1967) and Hawkins (2013), who writes on p. 326:

According to Miyake (1989, p. 347), Hasse introduced the term “Frobenius substitution” in (1926-1930), apparently unaware that Dedekind had discovered it independently and probably earlier.