The ability to evolve is a fundamental feature of biological systems, but the mechanisms underlying this capacity and the evolutionary dynamics of conserved core processes remain elusive. We show that yeast cells deleted of MYO1, encoding the only myosin II normally required for cytokinesis, rapidly evolved divergent pathways to restore growth and cytokinesis. The evolved cytokinesis phenotypes correlated with specific changes in the transcriptome. Polyploidy and aneuploidy were common genetic alterations in the best evolved strains, and aneuploidy could account for gene expression changes due directly to altered chromosome stoichiometry as well as to downstream effects. The phenotypic effect of aneuploidy could be recapitulated with increased copy numbers of specific regulatory genes in myo1Δ cells. These results demonstrate the evolvability of even a well-conserved process and suggest that changes in chromosome stoichiometry provide a source of heritable variation driving the emergence of adaptive phenotypes when the cell division machinery is strongly perturbed.