By leveraging the complementary advantages of free-space optical (FSO) and radio frequency (RF) links, hybrid FSO/RF systems exhibit broad application prospects. However, maintaining robustness while performing trade-off optimization between reliability and transmission efficiency under dynamic conditions with power constraints remains challenging. To address this, we propose a weather-aware multi-objective adaptive power allocation approach for hybrid FSO/RF systems based on the Non-dominated Sorting Genetic Algorithm II (NSGA-II) and a deep neural network (DNN). Closed-form expressions for the average bit-error rate (ABER) and average channel capacity (ACAP) are derived and used to evaluate NSGA-II objectives, generating a labeled optimal allocation dataset across diverse scenarios. A DNN is then trained on the dataset to learn adaptive power allocation strategies for dynamic environments. Numerical results demonstrate that the proposed scheme effectively achieves adaptive power allocation and significantly outperforms existing benchmark schemes. In dynamic scenarios, it reduces the ABER by 1–2 orders of magnitude and substantially lowers the outage probability (OP), while improving the overall ACAP by more than 0.5 Gbps under the same transmit power.