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We propose an accelerated failure time model with random effects for correlated or clustered survival time data. We assume that the error distribution belongs to the family of G-rho distribution and random effects follow multivariate normal distribution. The G-rho family distribution allows us to model ``attenuating'' or ``converging'' hazard functions over time, which represent a type of non-proportional hazards, with the special case of rho=1 corresponding to the proportional odds model. In G-rho family distribution, the larger the value of rho the higher the degree of non-proportionality in the data. Thus, estimating rho as a regression parameter instead of assuming a priori fixed value allows us more flexibility handling many different types of non-proportional converging hazards occurring in practice. We utilize EM algorithm for estimation. More specifically, Stochastic Expectation-Maximization (StEM) algorithm is used to maximize the complete data log-likelihood in the presence of random effects. The conditional expectation in the classical E-step is replaced by a stochastic draw from the posterior distribution of the latent variable via Gibbs sampler method. The computational complexity of the likelihood then can be avoided by maximizing the pseudo-complete data. We also examine the robustness of the estimated fixed effects and the estimated variance components when the error distribution or distribution of the random effects is misspecified through simulation studies, followed by an application to a clinical trial dataset. Public health significance: The proposed method enables researchers 1) to model dependent or clustered survival data which arise frequently in medical research; for example, in familial studies or multi-center clinical trials. 2) to model attenuating hazards which is a type of non-proportional hazard. 3) to reduce bias in estimating treatment effects in the presence of non-proportional hazards and unobserved heterogeneity. The proposed method contributes to more accurate estimation of important covariate effects (such as treatment effects) in practical settings such as in randomized clinical trial.