Radio relics are diffuse radio sources found in the outskirts of galaxy clusters and they are thought to trace synchrotron-emitting electrons accelerated via diffusive shock acceleration (DSA) at merger-driven shocks. For some radio relics, the shock Mach number inferred from the radio spectral index is often larger than that estimated from the discontinuities in X-ray observations. Here we study the re-acceleration model in which a weak shock encounters the intracluster plasma with a pre-existing population of relativistic electrons. Toward this end, we perform DSA simulations of spherical shocks with the parameters relevant for cluster radio relics and calculate the ensuing radio synchrotron emission from re-accelerated electrons. We demonstrate that the re-accelerated electron spectrum depends on the shape of the preshock electron spectrum, so the shock Mach number cannot be estimated directly from radio spectral index. Moreover, this scenario alleviates the problems regarding low acceleration efficiency of weak shocks and low frequency of merging clusters with detected radio relics.