Original usage of 'Bénabou cosmos' A (Bénabou) cosmos is a bicomplete closed symmetric monoidal category (see, for example, the $n$Lab).
However, I can't find the paper where Bénabou first uses this term - googling turns up nothing.
Does anybody know where it is first used, or how I could find out?
Edit: having read through all the summaries of papers by Bénabou that I could find, I couldn't find any references to the word cosmos.
This search is not going well...
 A: Like the other answers given so far, the following is not an answer to what the opening poster is really asking about (i.e., where Bénabou first published the word).
The following was kindly pointed out to me by someone else in an email I read today, and, lest I forget this topic because of other things, I'll quickly incorporate the email, lightly redacted, into an answer to the opening poster.
It seems to me, not knowing why, the most polite modus operandi to not acknowledge my correspondent, and trust that they'll let me know if they wish to be so acknowledged. Citing the email seems the right form of  acknowledgement here.

[A relevant reference] is Street's Elementary Cosmoi (https://link.springer.com/chapter/10.1007/BFb0063103), p. 134 (first page):
[Street writes:] "Our use of the word 'cosmos' is presumptuous. To Bénabou the word means 'bicomplete symmetric monoidal closed category', such categories V being rich enough so that the theory of categories enriched in V develops to a large extent as the theory of ordinary categories."
There are three papers (co)authored by Bénabou in the references; I'm not sure which one mentions the word.

I can offer a minuscule further piece of relevant information for this open question:

of the three references mentioned in [Street: Elementary cosmoi], the reference [J. Bénabou and J. Roubaud, Monades et descente. C.R. Acad. Sc. Paris 270 (1970) 96–98] is not where you'll find an answer: I translated this article here, and 'cosmos' does not appear therein.

So it seems it remains for you to read the following two:

*

*J. Bénabou, Introduction to bicategories. Lecture Notes in Mathematics 47 (1967) 1–77


*J. Bénabou, Catégories avec multiplication. C.R. Acad. Sc. Paris 256 (1963) 1887–1890
I haven't looked into those. Good luck for finding it.
A: It may well be that this object was named by another mathematician after Bénabou. There is a funny anecdote about Hilbert going to a conference and having to ask to a colleague what the heck were those Hilbert spaces everybody was talking about...
A: Although I haven't been able to track down the first usage of the word, I've heard the following etymological possibility from my supervisor: catégorie monoïdale symétrique gets initialised to CMS which gets pronounced cosmos.
