Recent years have seen significant interest in the mechanical properties of clay–polymer hybrids due to their suitability for possible application as sustainable materials in green chemistry. The objective of the present study was to investigate the mechanical properties of clay–polymer hybrids and their corresponding pristine smectite clay minerals. The density functional theory (DFT) method, employing the D3 scheme for corrections of dispersion interactions, was used to calculate elastic constants (Cij) of models of pristine smectites, particularly montmorillonite, beidellite, saponite, and hectorite, and their hybrids built on the polymer poly(2-methyl-2-oxazoline), PMeOx. Following that, the elastic moduli, encompassing the bulk modulus (KVRH), shear modulus (GVRH), Young’s modulus (EVRH), and Poisson’s ratio (ν), were calculated. The results revealed a reduction in elastic constants and elastic moduli following the intercalation of smectite clay minerals with the PMeOx polymer. The findings highlighted a distinctive ranking of mechanical properties among pristine smectite clay minerals and clay–polymer hybrids, with hectorite and its hybrid (Htr-PMeOx) demonstrating better performance compared with saponite, montmorillonite, and beidellite and their respective hybrids.