This answer concerns contramodules that are finitely generated as objects of the category of contramodules, in the sense of the abstract category-theoretic definition of "finitely generated".  The examples below in this answer tend to suggest that there are too few such finitely generated contramodules to hope for an interesting theory.

Let $\mathbf{K}$ be a cocomplete category (i.e., a category with set-indexed colimits).  An object $Q\in\mathbf{K}$ is said to be finitely generated if, for any diagram $(E_\gamma)_{\gamma\in\Gamma}$ of objects in $\mathbf{K}$, indexed by a directed poset $\Gamma$, with monomorphisms $E_\gamma\to E_\delta$ for $\gamma<\delta\in\Gamma$, the natural map
$$
 \varinjlim\nolimits^{\mathbf{Sets}}_{\gamma\in\Gamma}
 \operatorname{Hom}_{\mathbf{K}}(Q,E_\gamma)
 \longrightarrow
 \operatorname{Hom}_{\mathbf{K}}(Q,\,
 \varinjlim\nolimits^{\mathbf{K}}_{\gamma\in\Gamma}E_\gamma)
$$
is an isomorphism (bijection) of sets.

The following example of a direct limit is helpful to consider.  In any cocomplete category $\mathbf{K}$, for any family of elements $(P_x\in\mathbf{K})_{x\in X}$, the coproduct of all the objects $P_x$ in $\mathbf{K}$ is the direct limit of the coproducts of $(P_z)_{z\in Z}$ taken over all the finite subsets $Z\subset X$.
So let $\Gamma$ denote the directed poset of all finite subsets $Z\subset X$, ordered by inclusion.  Then we have
$$
 \coprod\nolimits^\mathbf{K}_{x\in X}P_x=
 \varinjlim\nolimits^{\mathbf K}_{Z\in\Gamma}
 \left(\coprod\nolimits^{\mathbf K}_{z\in Z}P_z\right).
$$
In particular, when all the objects $P_x$ are the same, $P_x=P$ for all $x\in X$, we have
$$
 P^{(X)} = \varinjlim\nolimits^{\mathbf K}_{Z\in\Gamma} P^{(Z)},
$$
where the notation is $P^{(Y)}=\coprod^{\mathbf K}_{y\in Y}P$.

If $\mathbf{K}$ is a category with zero object then, for any objects $E$ and $F\in\mathbf{K}$, the coproduct injections $E\to E\sqcup F$ and $F\to E\sqcup F$ are monomorphisms.  So in particular, in the above example, the $\Gamma$-indexed directed diagram of finite copowers $P^{(Z)}$ of an object $P$ is a diagram of monomorphisms.

Now let $C$ be a coassociative, counital coalgebra over a field $k$.
We are interested in the category of left $C$-contramodules $\mathbf{K}=C{-}\mathbf{contra}$.  Given a vector space $V$, the $C$-contramodule $F=\operatorname{Hom}_k(C,V)$ (with the contraaction map $\pi_F$ induced by the comultiplication map $\Delta_C$) is called the free $C$-contramodule generated by $V$.  Let $X$ be a set indexing a basis in $V$.

We will use the notation $A[X]=\bigoplus_{x\in X}^{\mathbf{Ab}}A$ for the direct sum of copies of the same object in the category of abelian groups (or vector spaces).  So, in particular, we have $V=k[X]$.  The $C$-contramodule $\operatorname{Hom}_k(C,k[X])$ can be also called the free $C$-contramodule generated by set $X$.  In particular, consider $P=C^*=\operatorname{Hom}_k(C,k)$; this is the free $C$-contramodule with one generator.  The notation $C^*[[X]]=\operatorname{Hom}_k(C,k[X])$ is sometimes used.

In the category of $C$-contramodules $\mathbf{K}=C{-}\mathbf{contra}$, we have
$$
 \operatorname{Hom}_k(C,k[X])=\coprod\nolimits_{x\in X}^{\mathbf K}C^*=P^{(X)}.
$$
Hence, following the above discussion of an arbitrary category $\mathbf{K}$,
$$
 \operatorname{Hom}_k(C,k[X])=
 \varinjlim\nolimits_{Z\in\Gamma}^{\mathbf K}
 \operatorname{Hom}_k(C,k[Z]),
$$
where $\Gamma$ is the directed poset of all finite subsets $Z\subset X$.

Now let $Q=C^*$ be also the free $C$-contramodule with one generator.  Then the functor $\operatorname{Hom}_{\mathbf K}(Q,{-})\colon C{-}\mathbf{contra}\to\mathbf{Sets}$ is simply the forgetful functor $C{-}\mathbf{contra}\to\mathbf{Sets}$ (assigning to a $C$-contramodule $M$ its underlying set $M$).

Hence we have
\begin{multline*}
 \varinjlim\nolimits^{\mathbf{Sets}}_{Z\in\Gamma}
 \operatorname{Hom}_{\mathbf{K}}(Q,P^{(Z)})
 = \varinjlim\nolimits^{\mathbf{Sets}}_{Z\in\Gamma}
 \operatorname{Hom}_k(C,k[Z]) \\
 = \varinjlim\nolimits^{\mathbf{Sets}}_{Z\in\Gamma}
 (C^*\otimes_k k[Z])
 = C^*\otimes_k k[X]=C^*[X].
\end{multline*}

On the other hand,
$$
 \operatorname{Hom}_{\mathbf{K}}(Q,\,
 \varinjlim\nolimits^{\mathbf{K}}_{Z\in\Gamma} P^{(Z)})
 = \operatorname{Hom}_{\mathbf{K}}(Q,P^{(X)})=
 \operatorname{Hom}_k(C,k[X])=C^*[[X]].
$$

Whenever the coalgebra $C$ is infinite-dimensional over $k$ and the set $X$ is infinite, the natural (injective) map $C^*\otimes_k k[X]\to\operatorname{Hom}_k(C,k[X])$ is not an isomorphism.  Thus, for any infinite-dimensional coalgebra $C$, the free $C$-contramodule with one generator $Q=C^*$ is not a finitely generated object in $\mathbf{K}=C{-}\mathbf{contra}$.

This is too long already, so I will perhaps continue with a specific counterexample in a comment.