Thermal storage units are pivotal to effective energy management, yet their efficiency hinges on minimizing thermal resistance during phase change processes. This study proposes a novel octagonal shell-and-tube thermal storage unit integrated with copper nanoparticle-enhanced phase change material (PCM) and three innovative fin configurations—straight dual-radial, tree-like, and root-like—to optimize heat transfer and accelerate melting. The root-like fin design, inspired by natural root morphology, features intricate branching to maximize the surface area and disrupt thermal stratification. Copper nanoparticles (0–8 vol%) are dispersed in the PCM to augment thermal conductivity. Using the Galerkin finite element method, the system is modeled to evaluate melting dynamics and thermal performance. The results reveal that root-like fins outperform conventional designs, reducing melting time by 56% compared to dual-radial fins and 91% relative to tree-like fins, owing to enhanced heat distribution and reduced thermal resistance. Additionally, an 8 vol% nanoparticle concentration improves thermal storage performance by 28.9% over pure PCM, accelerating energy absorption. The synergistic combination of biomimetic root-like fins and nanoparticle doping emerges as a transformative strategy for thermal storage systems, significantly enhancing energy discharge rates and operational efficiency. This study advocates adopting root-like fin architectures with copper nanoparticle-enhanced PCMs in applications such as solar energy storage and industrial waste heat recovery, offering a pathway toward sustainable and high-performance thermal management solutions.