To understand the structural stability, physical properties, and hardness of the Rh–Si system, we have performed systematic first-principles crystal structure searches for various stoichiometries of rhodium silicides, utilizing the particle swarm optimization method. A new stable stoichiometry, Rh₄Si₅ with space group P2₁/m, has been found at atmospheric pressure, complementing the three well-known rhodium silicides of Rh₂Si (Pnma), Rh₅Si₃ (Pbam), and RhSi (Pnma). Our calculations of the structural and mechanical properties of the known phases are in excellent agreement with the available experimental data and similar theoretical calculations. The elastic, electronic, and hardness properties of the Rh–Si stoichiometries are discussed. Our results suggest that the new rhodium silicide Rh₄Si₅ (P2₁/m) is a potentially hard material with the hardness of 20.1 GPa.