With advances in technology, hyperelastic materials are seeing use in varied applications ranging from microfluidic pumps, artificial muscles to deformable robots. Development of such complex devices is leading to increased use of hyperelastic materials in the construction of components undergoing dynamic excitation such as the wings of a micro-unmanned aerial vehicle or the body of a serpentine robot made of hyperelastic polymers. Since the strain energy potentials of various hyperelastic material models have nonlinearities present in them, exploration of their nonlinear dynamic response lends itself to some interesting consequences. In this work, a structure made of a Mooney–Rivlin hyperelastic material and undergoing planar vibrations is considered. Since the Mooney–Rivlin material's strain energy potential has quadratic nonlinearities, a possibility of 1:2 internal resonance is explored. A finite element method (FEM) formulation implemented in Matlab is used to iteratively modify a base structure to get its first two natural frequencies close to the 1:2 ratio. Once a topology of the structure is achieved, the linear mode shapes of the structure can be extracted from the finite element analysis, and a more complete nonlinear Lagrangian formulation of the hyperelastic structure can be used to develop a nonlinear two-mode dynamic model of the structure. The nonlinear response of the structure can be obtained by application of perturbation methods such as averaging on the two-mode model. It is shown that the nonlinear strain energy potential for the Mooney–Rivlin material makes it possible for internal resonance to occur in such structures. The effect of nonlinear material parameters on the dynamic response is investigated.