This paper sets up a rational inattention model for the route choice problem in a stochastic network where travelers face random travel time. Previous research has assumed that travelers incorporate all provided information without effort. This study assumes that information is costly and that travelers rationally choose how much information to acquire prior to choosing route. We begin with a single traveler and then extend the model to heterogeneous travelers where rationally inattentive user equilibrium (RIUE) is achieved. From the perspective of a single traveler, more information always reduces the impact of travel time variability and increases the probability of choosing a less costly route. However, in RIUE, more information may reduce the social welfare in some scenarios.