Route choice modeling under travel time uncertainty is essential for analyzing travelers’ mobility patterns. This paper investigates the impact of travel time uncertainty on route choice behavior in user equilibrium models based on a generalized mean-variance metric (GMV). This model can capture the influence of risk attitudes and schedule unpunctuality on route choice using a generalization of expected travel time, variance, and expected early or late arrival penalties, of which travelers are assumed to minimize the GMV of trips considering a certain on-time arrival probability. This paper establishes the properties of GMV and formulates the GMV-based static user equilibria as a variational inequality (VI) problem, for which the existence and uniqueness of the solutions are also analyzed. An effective traffic assignment algorithm without path enumeration is developed to solve the proposed user equilibrium problem. Numerical examples are conducted to demonstrate the properties of the proposed model.