Estimating the remaining useful life (RUL) of an asset lies at the heart of prognostics and health management (PHM) of many operations-critical industries such as aviation. Mod- ern methods of RUL estimation adopt techniques from deep learning (DL). However, most of these contemporary tech- niques deliver only single-point estimates for the RUL without reporting on the confidence of the prediction. This practice usually provides overly confident predictions that can have severe consequences in operational disruptions or even safety. To address this issue, we propose a technique for uncertainty quantification (UQ) based on Bayesian deep learning (BDL). The hyperparameters of the framework are tuned using a novel bi-objective Bayesian optimization method with objectives the predictive performance and predictive uncertainty. The method also integrates the data pre-processing steps into the hyperparameter optimization (HPO) stage, models the RUL as a Weibull distribution, and returns the survival curves of the monitored assets to allow informed decision-making. We vali- date this method on the widely used C-MAPSS dataset against a single-objective HPO baseline that aggregates the two ob- jectives through the harmonic mean (HM). We demonstrate the existence of trade-offs between the predictive performance and the predictive uncertainty and observe that the bi-objective HPO returns a larger number of hyperparameter configurations compared to the single-objective baseline. Furthermore, we see that with the proposed approach, it is possible to configure models for RUL estimation that exhibit better or comparable performance to the single-objective baseline when validated on the test sets.