This contribution presents a framework for calculating a sensitivity measure for problems of computational stochastic mechanics. More specifically, the sensitivity measure considered is the derivative of the failure probability with respect to parameters of the probability distributions (e.g. mean value, standard deviation) associated with the random input quantities of a system's model. The proposed framework is formulated as a post-processing step of Line Sampling, which is a simulation-based method for estimating small failure probabilities. In particular, the proposed framework comprises two different approaches for estimating the sought sensitivity. The application of the proposed framework and comparison of the two aforementioned approaches is discussed through a number of numerical examples. The results obtained indicate that both approaches allow estimating the sought sensitivity measure.