In this paper, we consider the problem of designing distributed control algorithms to solve the rendezvous problem for multi-robot systems with limited sensing, for situation in which random nodes may fail during execution. We first formulate a distributed solution based upon averaging algorithms that have been reported in the consensus literature. In this case, at each stage of execution a 1-step sequential optimal control (i.e., naïve greedy algorithm) is used. We show that by choosing an appropriate constraint set, finite-time point convergence is guaranteed. We then propose a distributed stochastic optimal control algorithm that minimizes a mean-variance cost function for each stage, given that the probability distribution for possible node failures is known a priori. We show via simulation results that our algorithm provides competitive rendezvous task performance in comparison to that of the classical circumcenter algorithm for cases in which there are no node failures. Then we show, via examples with multiple node failures, that our proposed algorithm provides better rendezvous task performance than contemporary algorithms in cases for which failures occur. Additionally, we generate and compare a spectrum of results by varying the probabilities of node failures, or varying the weight value for the variance term in the cost functional. The results suggest that by choosing the design parameters appropriately, one may adjust the degree of soft constraints of the controller as well.