By detailed first-principles calculations we show that the Fermi energy and the Rashba splitting in disordered ternary surface alloys Bi(x)Pb(y)Sb(1 - x - y)/Ag(111) can be independently tuned by choosing the concentrations x and y of Bi and Pb, respectively. The findings are explained by three fundamental mechanisms, namely the relaxation of the adatoms, the strength of the atomic spin-orbit coupling, and band filling. By mapping the Rashba characteristics, i.e. the splitting k(R) and the Rashba energy E(R), and the Fermi energy of the surface states in the complete range of concentrations, we find that these quantities depend monotonically on x and y, with a very few exceptions. Our results suggest that we should investigate experimentally effects which rely on the Rashba spin-orbit coupling depending on spin-orbit splitting and band filling.