Copper surfaces exhibit high catalytic selectivity but have poor hydrogen dissociation kinetics; therefore, we consider icosahedral Cu-13 nanoclusters to understand how nanoscale structure might improve catalytic prospects. We find that the spin state is a surprisingly important design consideration. Cu-13 clusters have large magnetic moments due to finite size and symmetry effects and exhibit magnetization-dependent catalytic behavior. The most favorable transition state for hydrogen dissociation has a lower activation energy than that on single-crystal copper surfaces but requires a magnetization switch from 5 to 3 mu(B). Without this switch, the activation energy is higher than that on single-crystal surfaces. Weak spin-orbit coupling hinders this switch, decreasing the kinetic rate of hydrogen dissociation by a factor of 16. We consider strategies to facilitate magnetization switches through optical excitations, substitution, charge states, and co-catalysts; these considerations demonstrate how control of magnetic properties could improve catalytic performance.