In this paper,
we investigate a hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac measures.
Under assumption that the domain is convex and the mesh is quasi-uniform, a priori error estimate for the error in $L^2$-norm
is proved. By duality argument and Oswald interpolation, a posteriori error estimates for the errors in $L^2$-norm and $W^{1,p}$-seminorm
are also obtained. Finally, numerical examples are provided to validate the theoretical analysis.