Stacks over the category locales are very interesting for topos theory:

A big success of topos theory is the fact that the $(2,1)$-categories of Grothendieck toposes and geometric morphisms between them embedded as a reflective full subcategory of the category of localic stacks, that is stack on the category of locales. It is in fact a full subcategory of the category of "Geometric localic stacks", that is those localic stacks comming from localic groupoids.

Note: There are some size issues involved whose discussion will be postponed to the very end.

We will identifies the category of locales with a full subcategory of the category of toposes, by identifying each locales $\mathcal{L}$ with the sheaf topos Sh$(\mathcal{L})$.

The basic idea is fairly simply given $\mathcal{T}$ a topos and $\mathcal{L}$ a locale, you get a category of geometric morphisms Hom$(\mathcal{L},\mathcal{T})$, if you simply drop the non-invertible natural transformations, you get a groupoid Hom$(\mathcal{L},\mathcal{T})$ of geometric morphisms and natural transformations.

This attach to every topos a pre-stack on the category of locales. It can be shown that this pre-stack is a stack for the topology whose coverings are the open surjections between locales (and the coproduct).

This construct a functor from the $(2,1)$-category of toposes to the $(2,1)$-category of localic stacks, which is fully faithful and identifies the category of toposes with a reflective full subcategory of stacks. Stack in the image are called "etale-complete" stack (to be honest one generally talks about étale-complete localic groupoids, but this is a property of the associated stack).

The starting point of this story started with the famous representation theorem of Joyal and Tierney in "An Extension of the Galois Theory of Grothendieck", which can be understood as the construction of the left adjoint, and the proof that it is essentially surjective, though most of the key idea are already present.

The results as presented above appeared in the two paper of Moerdijk:

The classying topos of a continuous groupoids, I & II

As the title suggest the results is mostly stated in terms of groupoids rather than stack, but the theory is really about stacks, and if I remember correctly the connection to stack is explicit mentioned in the paper. I think Bunge's paper "An application of descent to a classification theorem for toposes" is also relevant to the story.

So what I've said above is only correct up to some size consideration that need to be taken care of.

The category of locales, with the topology of open surjections, do not satisfies the smallness condition need in order for stackification to be well defined.

Though the point of view we adopt here, is that up to passing to a larger Grothendieck universe stackification is always defined, the question is only wether or not preserve it preserve certain smallness conditions.

In this case stackification do not preserve smallness: there are examples of small pre-stack of locales (in the sense "small colimits of representable") whose stackification is not even "levelwise small", that is $\mathcal{F}(\mathcal{L})$ can fail to be an essentially small groupoids.

But this is actually a good things, because for many Grothendieck topos, the groupoids Hom$(\mathcal{L},\mathcal{T})$ are not essentially small.

Here the appropriate "category of stack" to consider for what I say above to be correct are the large stack that are small colimits (in the category of stack) of representable stacks. This is not a locally small category (but the category of Grothendieck topos isn't either). The fact that the stack attached to a topos is in this category is non trivial, but follows directly from the work of Joyal and Tierney mentioned above.