The finite element method is commonly used to model tissue deformation in order to solve for unknown parameters in the inverse problem of viscoelasticity. Typically, a (regular-grid) structured mesh is used since the internal geometry of the domain to be identified is not known a priori. In this work, the generation of problem-specific meshes is studied and such meshes are shown to significantly improve inverse-problem elastic parameter reconstruction. Improved meshes are generated from axial strain images, which provide an approximation to the underlying structure, using an optimization-based mesh adaptation approach. Such strain-based adapted meshes fit the underlying geometry even at coarse mesh resolutions, therefore improving the effective resolution of the reconstruction at a given mesh size/complexity. Elasticity reconstructions are then performed iteratively using the reflective trust-region method for optimizing the fit between estimated and observed displacements. This approach is studied for Young's modulus reconstruction at various mesh resolutions through simulations, yielding 40%-72% decrease in root-mean-square reconstruction error and 4-52 times improvement in contrast-to-noise ratio in simulations of a numerical phantom with a circular inclusion. A noise study indicates that conventional structured meshes with no noise perform considerably worse than the proposed adapted meshes with noise levels up to 20% of the compression amplitude. A phantom study and preliminary in vivo results from a breast tumor case confirm the benefit of the proposed technique. Not only conventional axial strain images but also other elasticity approximations can be used to adapt meshes. This is demonstrated on images generated by combining axial strain and axial-shear strain, which enhances lateral image contrast in particular settings, consequently further improving mesh-adapted reconstructions.