An output-least-squares formulation for a class of parameter identification problems for elliptic variational inequalities is considered. Based on the concept of C-stationarity an active set type solver with feasibility restoration is introduced. It is shown that the new method relates to the so-called implicit programming techniques in the context of mathematical programs with equilibrium constraints. In the discrete setting, in order to overcome the ill-posedness of the problem, the parameter of interest is discretized on a coarser mesh than the state of the system. In addition, if the parameter corresponds to the coefficient in the bilinear form of the underlying differential operator, an interior-point treatment is employed to maintain the coercivity of the elliptic operator. Moreover, the computational domain for the coefficient depends on the measurement data. The paper ends with a report on numerical tests including an application to a simplified lubrication problem in a rolling element device.