There won't be any property which really distinguishes $\mathcal{O}_X$ inside $\mathsf{Mod}(X)$, since any invertible $\mathcal{O}_X$-module $\mathcal{L}$ induces an auto-equivalence of categories $\mathcal{L} \otimes -$. Instead, we may hope for properties of invertible $\mathcal{O}_X$-modules (which only have to be proven for the special case $\mathcal{O}_X$ as soon as they are categorical). I hope that you don't mind that I switch to $\mathsf{Qcoh}(X)$ when appropriate, because $\mathsf{Mod}(X)$ is too large and does not really incorporate the condition that $X$ is a scheme.
 
1) Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then every epimorphism $\mathcal{L} \to \mathcal{L}$ is an isomorphism.

2) The endomorphism ring $\mathrm{End}(\mathcal{L})$ is commutative. In particular, the group $\mathrm{Aut}(\mathcal{L})$ is commutative.
 
3) If $X$ is quasi-compact and quasi-separated, then $\mathcal{L}$ is a [finitely presentable object][1] of $\mathsf{Qcoh}(X)$.

4) If $X$ is separated, then every quasi-coherent $\mathcal{O}_X$-module is a subquotient of a direct sum of copies of $\mathcal{L}$. I don't know a classical reference for this (anyone?), but it is proven in Proposition 3.18 [here][2] (let $I=0$ there).


  [1]: https://ncatlab.org/nlab/show/compact+object
  [2]: https://arxiv.org/pdf/1310.5978v3.pdf