As far as I know the word *separable* was introduced by M. Fréchet in *Sur quelques points du calcul fonctionnel*, Rend. Circ. Mat. Palermo **22** (1906), 1-74. The paper can be obtained via <a href="http://dx.doi.org/10.1007/BF03018603">this link</a> (Springer). It's the famous paper in which he introduced metric spaces. He considers first slightly more general objects which he calls *classes (V)*:  where (V) stands for *voisinage* &mdash; neighborhood.

**Remark:** Metrics are introduced under the name *&eacute;cart* in n<sup>o</sup>&nbsp;49 on page&nbsp;30. It is peculiar that the symmetry condition is *not* explicitly mentioned but it seems to be understood as Fr&eacute;chet immediately mentions that metric spaces generalize *classes (V)* cf.&nbsp;n<sup>o</sup>&nbsp;27 on page&nbsp;17f. However, I couldn't find an instance where he actually uses it, he is always careful to respect the order&nbsp;&mdash;&nbsp;I may have missed something since I haven't read the paper in detail.

I quote the relevant passage [from n<sup>o</sup>&nbsp;37 on page 23f]:

>  Nous appellerons ensuite *classe s&eacute;parable* une classe qui puisse &ecirc;tre consid&eacute;r&eacute;e d'au moins une fa&ccedil;on comme l'ensemble d&eacute;riv&eacute; d'un ensemble d&eacute;nombrable de ses propres &eacute;l&eacute;ments.
>
> [...]
>
> Ceci &eacute;tant, nous nous bornerons maintenant &agrave; l'&eacute;tude des *classes (V)* NORMALES, *c'est-&agrave;-dire parfaites, s&eacute;parables et admettant une g&eacute;n&eacute;ralisation du th&eacute;or&egrave;me de* CAUCHY. Cette limitation n'a du reste rien d'artificiel, elle provient directement de la comparaison des classes (V) avec les ensembles lin&eacute;aires [...]
>
> [...]
>
> Passons maintenant aux classes s&eacute;parables. On peut qualifier ainsi les ensembles lin&eacute;aires en consid&eacute;rant la droite ind&eacute;finie comme l'ensemble d&eacute;riv&eacute; de l'ensemble des points d'abscisses rationnelles. Mais il n'en est pas de m&ecirc;me pour toute classe parfaite (V).


I am unable to translate this in a reasonable way (but see Amit's comment below for a translation). Very roughly: Fr&eacute;chet defines separable spaces as we do it today and says that in the following he will restrict attention to complete, perfect and separable metric spaces. The last quoted paragraph indeed confirms Qiaochu's comment.