System identification traditionally emphasizes modeling individual systems, which restricts the transferability of knowledge across related dynamics. Meta-learning provides a promising paradigm to overcome this limitation by leveraging prior experiences from diverse system behaviors, enabling more generalized and adaptive modeling. This paper investigates meta-model adaptation within in-context learning frameworks for system identification, with a focus on strategies to dynamically refine deep neural network-based meta-models. Specifically, we examine fine-tuning and learning-to-fine-tune approaches to efficiently adapt pre-trained meta-models to new system instances or evolving prediction tasks. The proposed methodology is validated through numerical experiments in three representative scenarios: specialization to a specific system, extension to previously unseen system classes, and recalibration for short- to long-term prediction tasks. Experimental results confirm substantial improvements in predictive accuracy, demonstrating that adaptation strategies enhance robustness and versatility. The findings highlight the potential of combining Transformer-based architectures with adaptive meta-learning to advance the application of intelligent system identification in domains such as robotics, process control, and dynamic cyber-physical systems.