A graph G is called (k,d) * -choosable if for every assignment L satisfying |L(v)|=k for all v V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. Let g(G) denote the girth of G and F the set of faces of G. In this paper, we prove the following results: for a graph G on surface with genus r>=2, we have: (a) G is (2,1) * -choosable if g(G)>=r+9 and |F|>=21; (b) G is (2,2) * -choosable if g(G)>= 65(r+5) and |F|>=13; (c) G is (2,3) * -choosable if g(G)>=r+5 and |F|>=14.