The idea which is being challenged, though certainly not disproved yet, is that there are new particles, other than the Higgs boson, that the LHC will be able to detect. It was very widely supposed that supersymmetric partners of some known particles would show up, because they could stabilize the mass of the Higgs boson.
The simplest framework for this is just to add supersymmetry to the standard model, and so most string models of the real world were built around this "minimal supersymmetric standard model" (MSSM). It's really the particle physicists who will decide whether the MSSM should lose its status as the leading idea for new physics. If they switch to some "new standard model", then the string theorists will switch too.
Whether they are aiming for the SM, the MSSM, or something else, the challenge for string theorists is, first, to find a shape for the extra dimensions which will make the strings behave roughly like the observed particles, and then second, use that model to predict something new. But as things stand, we still only have string models that qualitatively resemble reality.
Here is an example from a year ago - "Heterotic Line Bundle Standard Models". You'll see that the authors talk about constructing "standard models" within string theory. That means that the low-energy states in these string models resemble the particles of the standard model - with the same charges, symmetries, etc.
But that's still just the beginning. Then you have to check for finer details. In this paper they concern themselves with further properties like proton decay, the relative heaviness of the different particle generations, and neutrino masses. That already involves a lot of analysis. The ultimate test would be to calculate the exact masses and couplings predicted by a particular model, but that is still too hard for the current state of theory, and there's still work to do just in converging on a set of models which might be right.
So if supersymmetry doesn't show at the LHC, string theorists would change some of these intermediate criteria by which they judge the plausibility of a model, e.g. if particle physics opinion changed from expecting supersymmetry to show up at LHC energies, to expecting supersymmetry only to show up at the Planck scale. It would mean starting over on certain aspects of these model analyses, because now you have changed the details of your ultimate destination.