The category of elements of a presheaf $F:C^{\operatorname{op}} \to \operatorname{Set}$ is the most elementary form of the Grothendieck construction.
I think (but I am unable to find a reference) that the category of elements was studied before Grothendieck's definition for pseudofunctors with values in $\operatorname{Cat}$.
Maybe someone here who is more familiar with older papers in category theory can point to a reference.
Many thanks.
