We study area-stationary surfaces in the space $\mathbf{L}(\mathbf{H}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in $\mathbf{L}(\mathbf{H}^3)$ is an area-stationary surface. We then classify Lagrangian area-stationary surfaces $\Sigma$ in $\mathbf{L}(\mathbf{H}^3)$ and prove that the family of parallel surfaces in $\mathbf{H}^3$ orthogonal to the geodesics $\gamma\in \Sigma$ form a family of equidistant tubes around a geodesic. Finally we find an example of a two parameter family of rotationally symmetric area-stationary surfaces that are neither Lagrangian nor holomorphic.
