We present a new class of accelerated distributed algorithms for the robust solution of convex optimization problems over networks. The novelty of the approach lies in the introduction of distributed restarting mechanisms that coordinate the evolution of accelerated optimization dynamics with individual asynchronous and periodic time-varying momentum coefficients. We model the algorithms as set-valued hybrid dynamical systems since the method combines continuous-time dynamics with acceleration and set-valued discrete-time restarting updates. For these dynamics, we derive graph-dependent restarting conditions that guarantee suitable stability, robustness, and convergence properties in distributed optimization problems characterized by strongly convex primal functions. Our results are illustrated via numerical examples.