Errors in biomechanics simulations arise from modeling and discretization. Modeling errors are due to the choice of the mathematical model whilst discretization errors measure the impact of the choice of the numerical method on the accuracy of the approximated solution to this specific mathematical model. A major source of discretization errors is mesh generation from medical images, that remains one of the major bottlenecks in the development of reliable, accurate, automatic and efficient personalized, clinically-relevant Finite Element (FE) models in biomechanics. The impact of mesh quality and density on the accuracy of the FE solution can be quantified with a posteriori error estimates. Yet, to our knowledge, the relevance of such error estimates for practical biomechanics problems has seldom been addressed. In this contribution, we propose an implementation of some a posteriori error estimates to quantify the discretization errors and to optimize the mesh. More precisely, we focus on error estimation for a user-defined quantity of interest with the Dual Weighted Residual (DWR) technique. We test its applicability and relevance in two situations, corresponding to computations for a tongue and an artery, using a simplified setting, i.e., plane linearized elasticity with contractility of the soft-tissue modeled as a pre-stress. Our results demonstrate the feasibility of such methodology to estimate the actual solution errors and to reduce them economically through mesh refinement.