One possible answer to this could be Lang's theorem: it says that if $G/\mathbb{F}_q$ is a smooth connected algebraic group, then $H^1(\mathbb{F}_q,G)$ is trivial, or otherwise put every $G$-torsor has an $\mathbb{F}_q$-rational point. This generalizes your example: if $X/\mathbb{F}_q$ is a smooth projective variety such that $X_{\bar{\mathbb{F}}_q}$ is isomorphic to an abelian variety, then $X$ is a torsor under its Albanese variety $A = Alb(X)$. If $X$ has dimension $1$ then requiring the genus of $X$ to be $1$ is enough. If $X$ has dimension $2$ then by the classification of surfaces it is enough to assume that e.g. the canonical bundle of $X$ is trivial and $X$ is not simply connected.
Edit: as pointed out below, the previous sentence is correct only if the characteristic of $\mathbb{F}_q$ is $\neq 2,3$. If we additionally assume that the second $l$-adic Betti number equals $6$, then we get a criterion in all characteristics.