I'm sure the origin of the term is quite complex. Indeed, as Henry and David have pointed out, it is a very natural choice in this context.

I think the first *definition* of 'model' (not the first *use*) in the exact sense currently used in model theory is due to Tarski in *O pojȩciu wynikania logicznego* (*On the concept of following logically*; English translation [MR1951812](http://www.ams.org/mathscinet-getitem?mr=1951812); German [translation](http://gallica.bnf.fr/ark:/12148/bpt6k38370h/f12.image.r=.langFR) from Polish by Tarski himself). This is the paper where the current definition of logical consequence first appears:

> We say that the sentence $X$ *follows logically* from the sentences of the class
$\mathfrak{K}$ if and only if every model of the class $\mathfrak{K}$ is at the same time a model of the sentence $X$. [Translation by M. Stroinska and D. Hitchcock.]

I don't know much about Tarski's choice of terms here, but the commentary to the English translation by M. Stroinska and D. Hitchcock could be enlightening.

It is interesting to note that Tarski published his paper in 1936, half a decade after Gödel's *Die Vollständigkeit der Axiome des logischen Funktionenkalküls* (*The completeness of the axioms of the functional calculus of logic*; [MR1549799](http://www.ams.org/mathscinet-getitem?mr=1549799)). So it appears that these ideas were already known to members of the [Vienna Circle](http://en.wikipedia.org/wiki/Vienna_Circle) and affiliates.