This morning I introduced a notion of residual gerbe for a point x on an algebraic stack X. See the (currently) last section in the chapter Properties of Algebraic Stacks. I decided that the residual gerbe of X at x should be a reduced, locally Noetherian algebraic stack Z whose underlying space |Z| is a singleton which comes with a monomorphism Z —> X such that the unique point of Z maps to x.
In the generality of the stacks project I cannot show that residual gerbes always exist. If a residual gerbe Z does exist, then it is unique. In fact, it turns out that there exists a field and a surjective, flat, locally finitely presented morphism z : Spec(k) —> Z (which is a very convenient property to have because we work in the fppf topology). For any algebraic stacks there are alway points where the residual gerbe does exist, namely the points of finite type.
In Appendix B of the preprint “Etale devissage, descent and push-outs of algebraic stacks” David Rydh has shown (I think) that residual gerbes (as defined above) exist for any point of a quasi-separated algebraic stack (his results are actually stronger). This implies that the definition above does not conflict with the definition in the book by Laumon and Moret-Bailley.
One curiosity is that we haven’t yet defined gerbes in the stacks project. The fact that a residual gerbe “is” a gerbe (over something) isn’t yet documented in the stacks project.
Let me know if you have any comments or suggestions.