We study finite-time stability (FTS) for discrete-time systems with time-varying delay in this paper. Effect of nonlinear perturbations is also studied. Novel weighted summation inequalities, which generalize the discrete Jensen-based inequality, are established. These inequalities are used to analyze a Lyapunov-Krasovskii functional with a certain weight. New FTS criteria are derived to guarantee that the state of a system stays within a given threshold on a given interval. Numerical examples show that the new criteria can yield better results than others given in previous works.