We consider a cooperative beamforming system with the amplify-and-forward protocol. To reduce the signaling overhead, each relay quantizes the SNR of the source-to-relay (S-R) link into one bit, which is then transmitted through a binary symmetric channel (BSC) with a known crossover probability to the destination. Given the set of the error-corrupted one-bit messages received at the destination, the beamforming design criterion is the maximization of the expected receive SNR, averaged over the conditional bit-flipping distributions of BSCs. We derive an analytic expression for the considered SNR metric, which is a complicated function of the beamforming weights. To facilitate analysis, we further derive a tractable lower bound for the conditional average SNR. By conducting maximization with respect to this lower bound, a closed-form sub-optimal beamformer can be obtained as a solution to a generalized eigenvalue problem. Computer simulations are used to illustrate the performance of the proposed scheme.