Predicting tropical cyclone (TC) rapid intensification (RI) is an important yet challenging task in current operational forecast due to our incomplete understanding of TC nonlinear processes. This study examines the variability of RI onset, including the probability of RI occurrence and the timing of RI onset, using a low-order stochastic model for TC development. Defining RI onset time as the first hitting time in the model for a given subset in the TC-scale state space, we quantify the probability of the occurrence of RI onset and the distribution of the timing of RI onset for a range of initial conditions and model parameters. Based on asymptotic analysis for stochastic differential equations, our results show that RI onset occurs later, along with a larger variance of RI onset timing, for weaker vortex initial condition and stronger noise amplitude. In the small noise limit, RI onset probability approaches one and the RI onset timing has less uncertainty (i.e., a smaller variance), consistent with observation of TC development under idealized environment. Our theoretical results are verified against Monte-Carlo simulations and compared with explicit results for a general 1-dimensional system, thus providing new insights into the variability of RI onset and helping better quantify the uncertainties of RI variability for practical applications.