A detailed theoretical investigation on the structural, elastic, electronic, thermoelectric, thermodynamic and optical properties of half-Heusler MRhSb (M = Ti, Zr, Hf) compounds is presented. The computations are carried out using the full potential linear augmented plane wave method (FP-LAPW) within density functional theory (DFT). The optimized lattice parameters are in fairly good agreement with available experimental data. The computed elastic constants (C-ij) and their related elastic moduli confirm the stability of the studied compounds in the cubic phase and highlight their ductile nature. Analysis of band structures and densities of states (DOS) profiles reveal the semiconducting nature with an indirect energy band gap (Gamma-X). The bonding nature discussed via the electron charge density plot shows a mixture of covalent and ionic character. The evaluation of Seebeck coefficient leads to thermopower S >= 500 mu eV which is very benefic for thermoelectric applications. Estimated thermodynamic characteristic within the quasi-harmonic approximation shows similar behavior for the three compounds. Finally, some optical spectra such as the complex dielectric function, refractive index, reflectivity, energy loss function and absorption are presented.