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Recent developments in the field of computational modeling of fracture have opened up possibilities for designing structures against failure. A special case, called interfacial fracture or delamination, can occur in loaded composite structures where two or more materials are bonded together at comparatively weak interfaces. Due to the potential crack growth along these interfaces, the structural problem suffers from snap-back/snap-through instabilities and bifurcations with respect to the model parameters, leading to noisy and discontinuous responses. For such a case, the design optimization problem for a selected quantity of interest is ill-posed, since small variations in the design parameters can lead to large jumps in the structural response. To this end, this paper presents a stochastic optimization approach to maximize delamination resistance that is less sensitive to small perturbations of the design and thereby leads to a robust solution. To overcome the intractability of Monte Carlo methods for estimating the expected value of the expensive-to-evaluate response function, a global, piecewise-constant surrogate is constructed based on nearest-neighbor interpolation that is iteratively refined during the optimization run. We found that by taking a large stochastic region at the beginning of the optimization and gradually reducing it to the desired one can help overcome poor local optima. Our results demonstrate the effectiveness of the proposed framework using an example of shape optimization of hard inclusions embedded in a double-cantilever beam, which significantly enhances delamination resistance.