I'll use additive notation, and I'll assume that you are only considering finite cyclic groups. Let $\mathbf{Cyc}_p$ be the category of cyclic $p$-groups. Given $i,j\geq 0$ we can define $Q(p;i,j)\colon\mathbf{Cyc}\to\mathbf{Cyc}_p$ by $Q(p;i,j)(A)=\{a\in p^iA\colon p^ja=0\}$. We can also define a constant functor $C(p;i)\colon\mathbf{Cyc}\to\mathbf{Cyc}_p$ by $C(p;i)(A)=\mathbb{Z}/p^i$. Now suppose we have a collection of functors $F_p$, one for each prime $p$, each of the form $Q(p;i,j)$ or $C(p;i)$, and that only finitely many of the functors $F_p$ are constant. Then the group $F(A)=\prod_pF_p(A)$ is cyclic for all $A$, so we get a functor $F\colon\mathbf{Cyc}\to\mathbf{Cyc}$. I don't know if that gives all possible functors, but it certainly gives a reasonably rich supply of them.
As a very specific example, the functor $F(A)=(A/2A)\times(\mathbb{Z}/3)$ is non-constant with $F(0)\neq 0$.
UPDATE: Here's a more exotic example. If $X$ is a based set of size $1$ or $3$, there is a unique group structure for which the basepoint is the identity. If $A$ is cyclic of order $1$ or $7$ then we can impose an equivalence relation with $a\sim a^2\sim a^4$ for all $a$, and then $A/\sim$ has size $1$ or $3$ with basepoint $0$ and so has a group structure. This construction gives a functor on the category of groups of order $1$ or $7$, and we can compose with $A\mapsto A/7$ to get a functor $\mathbf{Cyc}\to\mathbf{Cyc}_3$. I am sure that there are many variations on this theme.