Here is what Peter Johnstone wrote on the matter in response to me passing on the question:

I was certainly around at the time that terminology came into use, and I may have been the first person to use it in print (in my 1977 book). In his 1975 paper "Change of base for toposes with generators", Radu Diaconescu referred to a topos F "having generators" over a base E, but that seemed a bit cumbersome; the single word "bounded" had the right overtones, in that it referred to a single object of F (which I later took to calling simply a "bound") that controlled how complicated the objects could be relative to those of E. I don't know whether I actually suggested the term, but as far as I can recall everyone agreed that it was a sensible choice.

Incidentally, the Grothendieck school were aware of the existence of what we would call unbounded Set-toposes, only they didn't consider them to be toposes. (There is one described in SGA4 as a "faux topos".)