What is the universal property of the Weyl group?

The group $WH$ is the automorphism group of $G/H$ as a $G$-set. Thus, $X^H$ is the set of morphisms from $G/H$ into $X$ in the category of $G$-sets, and this has an obvious action by precomposition with automorphisms.

If $F : C \to D$ is any functor whatsoever, the objects $F(c), c \in C$ always have a natural action of the automorphism group $\text{Aut}(F)$ of $F$ as a functor, and $\text{Aut}(F)$ is universal with respect to this property. If $F : C \to \text{Set}$ is a representable functor, this automorphism group coincides with the auotmorphism group of the representing object, by the Yoneda lemma. The functor of taking $H$-fixed points is representable by $G/H$, whose automorphism group is the Weyl group.

In general, it's an interesting question to ask what extra structure some objects $d \in D$ acquire by virtue of having been spit out by a functor $F : C \to D$. A very general answer to this question is that under mild hypotheses $F$ has what's called a codensity monad, and the objects $F(c), c \in C$ naturally acquire the structure of an algebra over this monad. If $F$ is the right adjoint of an adjunction this is the usual monad arising from that adjunction.

If $F : C \to \text{Set}$ is a representable functor represented by an object $c$, it has a left adjoint given by sending a set $X$ to the coproduct $\coprod_X c$ of $X$ copies of $c$ (assuming that these coproducts exist), and the codensity monad sends a set $X$ to the set $\text{Hom}(c, \coprod_X c)$. If $c$ is "connected" in the sense that a morphism $c \to \coprod_X c$ factors through one of the inclusions of a copy of $c$ (which is the case here, where $c = G/H$), then this is the monad whose algebras are $\text{End}(c)$-sets; otherwise it's more complicated.