Biomass-based functional polymers have received significant attention across various fields, in view of eco-friendly human society and sustainable growth. In this context, there are efforts to functionalize the biomass polymers for next-generation polymer materials. Here, stretchable heat transfer materials are focused on which are essential for stretchable electronics and future robotics. To achieve this goal, natural rubber (NR) is chemically modified with a thiol-terminated phenylnaphthalene (TTP), and then utilized as a thermally conductive NR (TCNR) matrix. Hexagonal boron nitride (h-BN), renowned for its high thermal conductivity and low electrical conductivity, is incorporated as a filler to develop stretchable heat transfer eco-materials. The optimized TCNR/h-BN composite elongates to 140% due to great elasticity of NR, and exhibits excellent dielectric properties (a low dielectric constant of 2.26 and a low dielectric loss of 0.006). Furthermore, synergetic phonon transfer of phenylnaphthalene crystallites and h-BN particles in the composite results in a high thermal conductivity of 0.87Wm-1K-1. The outstanding thermal, mechanical, and dielectric properties of the newly developed TCNR/h-BN composite enable the successful demonstration as stretchable and shape-adaptable thermal management materials.