$\def\sF{\mathcal{F}}$This is a partial answer, but anyway it might be useful.

Definition. We say that that a sheaf $\sF$ over $X$ satifies the identity principle if for every open and connected subset $U\subset X$ it holds that if $s,t\in\sF(U)$ are sections such that $s_x=t_x$ for some $x\in U$, then $s=t$.

If $\sF$ is a sheaf whose étale space is Hausdorff, then it must satisfy the identity principle: for an open and connected subset $U\subset X$, one considers the set $$ S=\{x\in U\mid s_x=t_x\}. $$ Then $S$ is open and, using that $|\sF|$ is Hausdorff, $S$ is closed. So $S\neq\varnothing$ is equivalent to $S=U$.

Thus, any candidate for a possible characterization of the Hausdorff condition on the étale space must imply the identity principle.

One can show that if $X$ is locally connected Hausdorff then the converse is also true ($\sF$ satifies id. principle implies $|\sF|$ Hausdorff). This is actually how one proves the Hausdorff condition on the étale space of the sheaf of holomorphic functions on $\mathbb{C}^n$.

$\def\sF{\mathcal{F}}$This is a partial answer, coming from Analysis. It didn't fit as a comment and nobody had pointed it out before, so I am posting it as an answer.

Definition. We say that that a sheaf $\sF$ over $X$ satifies the identity principle if for every open and connected subset $U\subset X$ it holds that if $s,t\in\sF(U)$ are sections such that $s_x=t_x$ for some $x\in U$, then $s=t$.

If $\sF$ is a sheaf whose étale space is Hausdorff, then it must satisfy the identity principle: for an open and connected subset $U\subset X$, one considers the set $$ S=\{x\in U\mid s_x=t_x\}. $$ Then $S$ is open and, using that $|\sF|$ is Hausdorff, $S$ is closed. So $S\neq\varnothing$ is equivalent to $S=U$.

Thus, any candidate for a possible characterization of the Hausdorff condition on the étale space must imply the identity principle.

One can show that if $X$ is locally connected Hausdorff then the converse is also true ($\sF$ satifies id. principle implies $|\sF|$ Hausdorff). This is actually how one proves the Hausdorff condition on the étale space of the sheaf of holomorphic functions on $\mathbb{C}^n$.