One aspect of Stone duality is the representation of each Boolean algebra as a subset B -> P(X) (for some set X) which is closed under the set-theoretic Boolean operations on P(X). My guess is that this result persists over an arbitrary Boolean topos E, provided one assumes there are enough prime ideals, ie. that for each Boolean algebra B over V, the subobject X -> E(B,Omega) consisting of lattice homomorphisms is (in some sense) large enough??? Cheers, PBJ.