DNA-based data storage offers an attractive alternative to traditional media due to its exceptional density, durability, and sustainability. However, errors introduced across the DNA storage pipeline critically impede accurate sequence reconstruction from noisy sequencing reads. This paper addresses the DNA sequence reconstruction problem by proposing FedDNA, a novel Personalized Federated Learning (PFL) framework based on Evidential Deep Learning (DEL), designed for DNA storage environments. FedDNA quantifies robust predictive uncertainty through a novel evidence fusion mechanism that aggregates evidence from each noisy read in a cluster, thereby enhancing client-level prediction reliability. For efficient sequence modeling and reconstruction from these noisy clusters, its architecture employs a convolution-enhanced Mamba encoder and an LSTM decoder. To address prohibitive centralized training costs, privacy concerns, and data heterogeneity across diverse DNA storage data, FedDNA integrates PFL and designs an innovative uncertainty-driven personalized aggregation strategy based on epistemic and aleatoric decomposition, for which we also provide rigorous theoretical generalization bounds. Experimental results demonstrate FedDNA achieves superior reconstruction performance on DNA storage data with heterogeneity, highlighting its potential for secure and efficient DNA storage systems.