A mesh with planar faces is called an edge offset (EO) mesh if there exists a combinatorially equivalent mesh such that corresponding edges of and lie on parallel lines of constant distance d. The edges emanating from a vertex of lie on a right circular cone. Viewing as set of these vertex cones, we show that the image of under any Laguerre transformation is again an EO mesh. As a generalization of this result, it is proved that the cyclographic mapping transforms any EO mesh in a hyperplane of Minkowksi 4-space into a pair of Euclidean EO meshes. This result leads to a derivation of EO meshes which are discrete versions of Laguerre minimal surfaces. Laguerre minimal EO meshes can also be constructed directly from certain pairs of Koebe meshes with help of a discrete Laguerre geometric counterpart of the classical Christoffel duality.