An analytical model of a single-atom electron source is presented, where electrons are created by near-threshold photoionization of an isolated atom. The model considers the classical dynamics of the electron just after the photon absorption, i.e. its motion in the potential of a singly charged ion and a uniform electric field used for acceleration. From closed expressions for the asymptotic transverse electron velocities and trajectories, the effective source temperature and the effective source size can be calculated. The influence of the acceleration field strength and the ionization laser energy on these properties has been studied. With this model, a single-atom electron source with the optimum electron beam properties can be designed. Furthermore, we show that the model is also applicable to ionization of rubidium atoms, thus also describes the ultracold electron source, which is based on photoionization of laser-cooled alkali atoms.