The author studies surfaces of revolution in the \(3\)-dimensional Minkowski space \({\mathbb E}^3_1\). It is proven that such surfaces have pointwise \(1\)-type Gauss map if and only if they have constant mean curvature. Here, a surface is said to have pointwise \(1\)-type Gauss map if \[ \Delta G=f\,G, \] where \(G\) is the Gauss map, \(\Delta\) the Laplacian and \(f\) a smooth function.