Here is a direct link to Nolte's Tangled Tale of Phase Space on Physics Today. Big takeaways: the name did not come from Liouville's oft-cited 1838 paper, and Boltzmann used "phase" without "space" in the right context back in 1872, and he is the one who fully developed the concept, with a big help from Jacobi's 1842-43 work.
Nolte also suggests that the combination "phase space" originates in "a now obscure encyclopedia article published in 1911 that had etymological side effects not fully intended by its authors", referring to Ehrenfests' article for the Encyclopedia of Mathematical Sciences written at Klein's request.
However, as @ConsigliereZARF pointed out in the comment, German Phasenraum appears earlier in two reviews of Gibbs's Elementary Principles in Statistical Mechanics (1902). One by Abraham in Physikalische Zeitschrift, v. 3, p. 582, and the other by Planck in Annalen der Physik und Chemie, v. 27, p.748. Although Gibbs himself does not quite form the combination "phase space", he writes the following in a footnote on p.11:
"If we regard a phase as represented by a point in space of $2n$ dimensions, the changes which take place in the course of time in our ensemble of systems will be represented by a current in such space. This current will be steady so long as the external coordinates are not varied."
He later analogizes the flow in the "space of $2n$ dimensions" to that of incompressible fluid. From there the shorthand of "phase space" comes naturally. In a different context, Phasenraum is mentioned in passing in a short note Rauchkeilbeobachtungen von η Aquilae by Plassman back in 1895.
Here is Nolte's account:
"In fact, in his paper Liouville makes no mention of phase space, let alone dynamical systems. Liouville’s paper is purely mathematical, on the behavior of a class of solutions to a specific kind of differential equation. Though he lived for another 44 years, he was apparently unaware of his work’s application to statistical mechanics by others even within his lifetime... Explicitly referencing Liouville’s 1838 paper in the 1866 publication of his lectures of 1842-43, Jacobi was the first to put Liouville’s mathematical theorem into a mechanical context. What Jacobi did not do, and indeed could not do in his time, was to represent mechanical systems within a generalized space.
[...] In 1872 Boltzmann used the term “phase” for the first time in a paper on the further studies of the equipartition theory of gas molecules... Boltzmann made the distinction between kind of motion (Bewegungsart, such as translational and rotational motion, which contribute to the total energy) and the phase of the motion (Bewegungsphase, such as the changing coordinate and momentum values of the motion). That is the defining moment for the word “phase” in phase space... Boltzmann, for his part, did not use the term “phase” after his 1872 paper until the publication of his Vorlesungen über Gastheorie (Lectures on Gas Theory) in 1896.
[...] Why does Liouville get the credit when it was Jacobi and Boltzmann who invented phase space and discovered conserved volumes in it? The answer is that Boltzmann himself gives Liouville the credit in his Lectures. Although Boltzmann had known, at the time of his early papers, of Jacobi’s reference to Liouville’s theorem, it was only later in his Lectures of 1896 that Boltzmann first placed Liouville’s name on the conservation theorem in a way that stuck. Had it not been for Jacobi’s reference to Liouville in his Lectures, Boltzmann likely never would have known of Liouville’s paper. In turn, had Boltzmann not given the credit to Liouville, then the conservation of phase-space volume could very reasonably have been called Boltzmann’s theorem. Ironically, by naming it “Liouville’s Theorem” Boltzmann obscured his own role in the discovery and use of phase space."