Let $X$ be a Riemann surface of genus $g > 0$. Let $S$ denote the set of local systems (locally constant sheaves) on $X$ with fiber $\mathbb{C}$. $S$ is in natural bijection with $H^1(X, \underline{\mathbb{C}^{\times}})) \cong (\mathbb{C}^{\times})^{2g}$, where $\underline{\mathbb{C}^{\times}}$ is the constant sheaf on $X$ with fiber $\mathbb{C}^{\times}$.

By the Riemann-Hilbert correspondence, each local system $\mathbb{L} \in S$ corresponds to a line bundle with a flat connection. This map is given by $\mathbb{L} \mapsto (\mathbb{L} \otimes_{\mathbb{C}} \mathcal{O}_X, \nabla = 1 \otimes d)$, where $d$ is the usual exterior derivative. Forgetting the connection gives a map from $S$ to the Picard group of $X$, $H^1(X, \mathcal{O}_X)$. This is the natural map $H^1(X, \underline{\mathbb{C}}^{\times}) \to H^1(X, \mathcal{O}_X^{*})$, so in particular it is a map of abelian groups.

What is the description of this map? In particular, I do not know how to show it has non-trivial image, i.e., that there is a non-trivial holomorphic line bundle that admits a flat connection.

It follows from Chern-Weil theory that the image is contained in the Jacobian (as the curvature of a connection is essentially the first Chern class). The identification of $S$ with $(\mathbb{C}^{\times})^{2g}$ is non-canonical, but the topology this gives to $S$ is canonical. I think one can show that this map is continuous by thinking about cocycles, which gives another proof that the image is in the Jacobian.