In Vakil's Foundations of Algebraic Geometry he defines the skyscraper sheaf as follows.
Let $X$ be a topological space with $p\in X$. Let $S$ be a set and $\{e\}$ any singleton set. Let $i_p:p\to X$ be the inclusion then define: $$ i_{p,_{*}S}(U) = \begin{cases} S,& p\in U\\ \{e\},& p\not\in U \end{cases} $$
My question is, is there any reason why we use a single-element set $\{e\}$? As far as I can tell, we still get a sheaf if we use an arbitrary set in place of $\{e\}$. Why don't we use, say, the empty set instead?
There's a simple pragmatic reason that you can't use the empty set in place of $\{e\}$, at least when $S$ is nonempty: If $i_{p,*}S(X) = S$, then for any open set $U$, there is a restriction map $\rho^X_U\colon S\to i_{p,*}S(U)$, so $i_{p,*}S(U)$ cannot be empty.
As for why the definition is as it is, the skyscraper sheaf satisfies a universal property: The stalk of $i_{p,*}X$ at $p$ is $S$, and for any other sheaf $F$ on $X$ such that $F_p = S$, there is a unique sheaf morphism $F\to i_{p,*}S$ such that the induced map $F_p \to (i_{p,*}S)_p$ is the identity map on $S$. That is, the skyscraper sheaf is terminal in the category of sheaves on $X$ with stalk $S$ at $p$ (where the morphisms are those which induce the identity map on $S$). The appearance of the singleton set $\{e\}$ can be explained by the fact that $\{e\}$ is the terminal object in $\mathsf{Set}$.
This universal property is a bit awkward to state, since the uniqueness of the map $F\to i_{p,*}S$ depends on an identification of the stalk of $F$ at $p$ with $S$. It's cleaner to view it as a special case of the fact (it's a good exercise, and I'd be surprised if it's not in Vakil's notes) that the skyscraper sheaf at $p$ functor $i_{p,*}\colon \mathsf{Set}\to \mathsf{Sh}(X)$ is right adjoint to to the stalk at $p$ functor $(-)_p\colon \mathsf{Sh}(X)\to \mathsf{Set}$. Then the canonical map in the previous paragraph is the image of the identity map on $S$ under the natural isomorphism $\mathrm{Hom}_{\mathsf{Set}}(F_p,S)\to \mathrm{Hom}_{\mathsf{Sh}(X)}(F,i_{p,*}S)$. To put it another way, it's the component $\eta_F$ at $F$ of the unit of this adjunction $\eta\colon \text{id}_{\mathsf{Sh}(X)} \to i_{p,*}(-)_p$.
This answer was in terms of sheaves of sets, but the same things are true for sheaves of abelian groups, modules, etc. as mentioned in Qiaochu's comment.