Let $(V,\xi)$ be a toric contact manifold of Reeb type that is a prequantization of a toric symplectic manifold $(M,\omega)$. A contact blow up of $(V,\xi)$ is the prequantization of a symplectic blow up of $(M,\omega)$. Thus, a contact blow up of $(V,\xi)$ is a new toric contact manifold of Reeb type. In some special cases we are able to compute the cylindrical contact homology for the contact blow-up using only the cylindrical contact homology of the contact manifold we started with.