We studied the dynamics of entropic uncertainty in Markovian and non-Markovian systems during the charging of open quantum batteries (QBs) mediated by a common dissipation environment. In the non-Markovian regime, the battery is almost fully charged efficiently, and the strong non-Markovian property is beneficial for improving the charging power. In addition, the results show that the energy storage is closely related to the couplings of the charger-reservoir and battery-reservoir; that is, the stronger coupling of a charger reservoir improves energy storage. In particular, entanglement is required to obtain the most stored energy and is accompanied by the least tight entropic bound. Interestingly, it was found that the tightness of the entropic bound can be considered as a good indicator of the energy transfer in different charging processes, and the complete energy transfer always corresponds to the tightest entropic bound. Our results provide insight into the optimal charging efficiency of QBs during practical charging.