Predicting the effect of a single amino acid substitution on the stability of a protein structure is a fundamental task in macromolecular modeling. It has relevance to drug design and understanding of disease-causing protein variants. We present KINARI-Mutagen, a web server for performing in silico mutation experiments on protein structures from the Protein Data Bank. Our rigidity-theoretical approach permits fast evaluation of the effects of mutations that may not be easy to perform in vitro, because it is not always possible to express a protein with a specific amino acid substitution. We use KINARI-Mutagen to identify critical residues, and we show that our predictions correlate with destabilizing mutations to glycine. In two in-depth case studies we show that the mutated residues identified by KINARI-Mutagen as critical correlate with experimental data, and would not have been identified by other methods such as Solvent Accessible Surface Area measurements or residue ranking by contributions to stabilizing interactions. We also generate 48 mutants for 14 proteins, and compare our rigidity-based results against experimental mutation stability data. KINARI-Mutagen is available at http://kinari.cs.umass.edu.