Determining the impact of mutations on the thermodynamic stability of proteins is essential for a wide series of applications such as rational protein design and genetic variant interpretation. Since protein stability is a major driver of evolution, evolutionary data are often used to guide stability predictions. Many state-of-the-art stability predictors extract evolutionary information from multiple sequence alignments (MSA) of proteins homologous to a query protein, and leverage it to predict the effects of mutations on protein stability. To evaluate the power of such methods and their limitations, we used the massive amount of stability data recently obtained by deep mutational scanning to study how best to construct MSAs and optimally extract evolutionary information from them. The parameters considered include the protein sequence dataset used for the homologous search, as well as MSA depth, E-value and curation criteria. We also tested different evolutionary models and unexpectedly found that independent-site models achieve the similar accuracy as more complex epistatic models. Interestingly, by combining any of the evolutionary features with a simple structural feature, the relative solvent accessibility of the mutated residue, we obtained similar prediction accuracy of supervised, machine learning-based, protein stability change predictors. Our results provide new insights into the relationship between protein evolution and stability, and show how evolutionary information can be exploited to improve the performance of mutational stability prediction.