Robust design optimization of stochastic black-box functions is a challenging task in engineering practice. Crashworthiness optimization qualifies as such problem especially with regards to the high computational costs. Moreover, in early design phases, there may be significant uncertainty about the numerical model parameters. Therefore, this paper proposes an adaptive surrogate-based strategy for robust design optimization of noise-contaminated models under lack-of-knowledge uncertainty. This approach is a significant extension to the robustness under lack-of-knowledge method (RULOK) previously introduced by the authors, which was limited to noise-free models. In this work, it is proposed to use a Gaussian Process as a regression model based on a noisy kernel. The learning process is adapted to account for noise variance either imposed and known or empirically learned as part of the learning process. The method is demonstrated on three analytical benchmarks and one engineering crashworthiness optimization problem. In the case studies, multiple ways of determining the noise kernel are investigated: (1) based on a coefficient of variation, (2) calibration in the Gaussian Process model, (3) based on engineering judgment, including a study of the sensitivity of the result with respect to these parameters. The results highlight that the proposed method is able to efficiently identify a robust design point even with extremely limited or biased prior knowledge about the noise.