@kimchi lover, gave the answer. The article:
Brush, Stephen G. "Boltzmann's “Eta Theorem”: Where's the Evidence?." American Journal of Physics 35.9 (1967): 892-892.
talks about the problem of the naming of the theorem. So initially Boltzmann used $E$ for entropy, but according to Sidney Chapman, S.H. Burbury changed the variable to $H$ (about 1890) and it stayed that way, even for Boltzmann. But up to 1963, nobody has come with a proof that the $H$ theorem is an capital Greek letter $\eta$.
I do not know now if in general the French physicists, or just some, call it 'eta' theorem, but I realize this seems just a generalized misnaming.
Update: I have found the following reference:
Hjalmars, Stig. "Evidence for Boltzmann’s H as a capital eta." American Journal of Physics 45.2 (1977): 214-215.
That hints that by the typographical styles of Greek and Latin letters in Boltzmann papers, and by the generalization of Gibbs from $\eta$ and $\psi$ to $H$ and $\Psi$ for negative entropy and free energy, the interpretation of "eta theorem" should be favored.