A set of edges $Γ$ of a graph $G$ is an edge dominating set if every edge of $G$ intersects at least one edge of $Γ$, and the edge domination number $γ_e(G)$ is the smallest size of an edge dominating set. Expanding on work of Laskar and Wallis, we study $γ_e(G)$ for graphs $G$ which are the incidence graph of some incidence structure $D$, with an emphasis on the case when $D$ is a symmetric design. In particular, we show in this latter case that determining $γ_e(G)$ is equivalent to determining the largest size of certain incidence-free sets of $D$. Throughout, we employ a variety of combinatorial, probabilistic and geometric techniques, supplemented with tools from spectral graph theory.