This paper develops a two-dimensional hyperchaotic discrete dynamical system (2D-HDDS) by extending the exponential-Chebyshev mechanism to a two-dimensional setting through a structured nonlinear feedback coupling. The resulting system combines the stretching effect of exponential dynamics with the folding behavior of Chebyshev mappings. Bifurcation diagrams and Lyapunov-exponent spectra indicate that 2D-HDDS exhibits a comparatively wide hyperchaotic parameter region and high dynamical complexity. Based on the generated keystream, we further construct an image-encryption scheme that integrates dual permutations, a key-dependent dynamic S-box, index-coupled forward and backward nonlinear diffusion, and a lightweight avalanche booster to enhance the propagation of plaintext perturbations. Experimental results show high key sensitivity and strong plaintext sensitivity, with the number of pixels change rate (NPCR) and the unified average changing intensity (UACI) values close to the ideal levels predicted by the standard random-like cipher-image model, while also demonstrating good resistance to statistical and differential attacks.