Flexible classes of survival models are proposed that naturally deal with both outlying observations and unobserved heterogeneity. We present the family of Rate Mixtures of Weibull distributions, for which a random effect is introduced through the rate parameter. This family contains the well-known Lomax distribution and can accommodate flexible hazard functions. Covariates are introduced through an Accelerated Failure Time model and we explicitly take censoring into account. We construct a weakly informative prior that combines the structure of the Jeffreys prior with a proper prior on the parameters of the mixing distribution, which is induced through a prior on the coefficient of variation. This improper prior is shown to lead to a proper posterior distribution under mild conditions. The mixing structure is exploited in order to provide an outlier detection method. Our methods are illustrated using two real datasets, one concerning bone marrow transplants and another on cerebral palsy.