In this work, the stiffness, i.e., the derivative of the load-separation curve, is studied for self-affine fractal surfaces with non-Gaussian height distribution. In particular, the heights of the surfaces are assumed to follow a Weibull distribution. We find that a linear relation between stiffness and load, well established for Gaussian surfaces, is not obtained in this case. Instead, a power law, which can be motivated by dimensionality analysis, is a better descriptor. Also unlike Gaussian surfaces, we find that the stiffness curve is no longer independent of the Hurst exponent in this case. We carefully asses the possible convergence errors to ensure that our conclusions are not affected by them.