A module $M$ such that $\mathrm{Hom}(M,-)$ preserves (infinite) direct sums is called dually slender. A ring is called steady if every dually slender module over it is finitely generated. A google research shows that in the last 15 years a lot of work has been devoted to the study of dually slender modules and steady rings, in particular by Jan Trlifaj and Jan Zemlicka. See also the comprehensive list of references in: Bashir, Kepka, Němec "Modules commuting (via Hom) with some colimits." (online).
An important characterization of dually slender modules is the following:
$M$ is dually slender iff for every chain of submodules $M_1 \subseteq M_2 \subseteq ...$ whose union is $M$, there is some $n$ with $M = M_n$.
You can find the proof as Lemma 1.1 in: Jan Zemlicka, "Class of dually slender modules" (online). In the introduction to Jan Zemlikca, "Steadiness of regular semiartinian rings with primitive factors artinian" (online) it is noted that three constructions of non-finitely generated dually slender modules are known - many references are given. An explicit example is finally given in Jan Zemlikca, "$\omega_1$-generated uniserial modules over chain rings" (online), Example 2.7:
Take the reverse of the natural order on the ordinal number $\omega_1$ and consider the lexicographic order on $\mathbb{Z}^{(\omega_1)}$. Pick a valuation domain $R$ whose value group is $\mathbb{Z}^{(\omega_1)}$. Then $R$ is not steady. In fact, this follows from a more general result (Corollary 2.6) which says that (a transfinite version of) the Krull dimension of steady chain rings is countable.