Amine-terminated butadiene-acrylonitrile (ATBN) was covalently grafted on a hexagonal boron nitride (BN) surface to enhance the interface affinity between BN and a carboxylated styrene-butadiene rubber (XSBR) matrix for preparing thermal interface materials with robust thermo-physical properties. The strong attraction between the amino groups on the ATBN-functionalized BN (ABN) and the carboxyl groups on the XSBR chains can further enhance the interfacial adhesion between the fillers and polymer matrix. The mechanical properties, thermal stability, thermal conductivity, thermo-physical properties, and interfacial filler–matrix network of the prepared XSBR/ABN nanocomposites were studied. The XSBR/ABN nanocomposites exhibited better mechanical and thermo-physical behavior compared to the XSBR/BN nanocomposites over the entire range of filler concentration, representing an enhanced interfacial interaction owing to functionalization. Moreover, the XSBR/ABN nanocomposites exhibited superior thermal conductivity as compared to the XSBR/BN nanocomposites, with the former exhibiting approximately 490% higher thermal conductivity than neat XSBR, whereas the latter exhibited 395% higher thermal conductivity than neat XSBR at a filler loading of 90 phr.