We develop case deletion diagnostics for prediction of future observations in the accelerated failure time model. We view prediction to be an important inferential goal in a survival analysis and thus it is important to identify whether particular observations may be influencing the quality of predictions. We use the Kullback-Leibler divergence as a measure of the discrepancy between the estimated probability distributions for the full and the case-deleted samples. In particular, we focus on the effect of case deletion on estimated survival curves but where we regard the survival curve estimate as a vehicle for prediction. We also develop a diagnostic for assessing the effect of case deletion on inferences for the median time to failure. The estimated median can be used with both predictive and estimative purposes in mind. We also discuss the relationship between our suggested measures and the corresponding Cook distance measure, which was designed with the goal of assessing estimative influence. Several applications of the proposed diagnostics are presented.