Rocks are known to fail in a time-dependent manner similar to many brittle and quasi-brittle materials. However, observed time dependence of hydraulic fracture initiation has ubiquitously been attributed to fluid flow-related mechanisms without consideration of the intrinsic time dependence of rock failure. Laboratory delayed hydraulic fracturing breakdown experiments on three rocks (granite, sandstone, and limestone) show sensitivity to rock properties associated with subcritical crack growth as well as fluid viscosity and ambient confining pressure applied to the specimens. A new hydraulic fracture initiation model accounting for subcritical crack growth and fluid flow in a poroelastic medium, as well as the additional energy dissipation required for fluid flow in the rough fracture tip, is then used to estimate the value of the subcritical index n based on characterization experiments. Given this characterization data, the model is shown to be capable of predicting time-dependent hydraulic fracture initiation with variation of fluid viscosity and confining stress.