The gelation kinetics of silica nanoparticles is a central process in physical chemistry, yet it is not fully understood. Gelation times are measured to increase by over 4 orders of magnitude, simply changing the monovalent salt species from CsCl to LiCl. This striking effect has no microscopic explanation within current paradigms. The trend is consistent with the Hofmeister series, pointing to short-ranged solvation effects not included in the standard colloidal (DLVO) interaction potential. By implementing a simple form for short-range repulsion within a model that relates the gelation timescale to the colloidal interaction forces, we are able to explain the many orders of magnitude difference in the gelation times at fixed salt concentration. The model allows us to estimate the magnitude of the non-DLVO hydration forces, which dominate the interparticle interactions on the length scale of the hydrated ion diameter. This opens the possibility of finely tuning the gelation time scale of nanoparticles by just adjusting the background electrolyte species.