Spin properties of single-doped and single-electron charged nano-systems having an odd number of electrons are studied. Starting from an expression for quasiparticle energies in the GW approximation, a simple analytical expression for the spin-splitting of an electron spectrum in such system is derived. First-principles calculations by the DFT-GGA, Hartree-Fock, GW- and hybrid functional methods, which were performed for the silicon clusters and metal phthalocyanine molecules of 1 nm diameter, support this analytical consideration. They show that the spin-splitting energy calculated by the DFT-GGA method is about one order lower, than the results obtained with the methods based on the many-electron theory. A large value of spin-splitting in investigated nano-systems, which is typically of several eV, has an origin in strong localization of electrons and weak screening of exchange interaction. A possible use of this effect in spintronic applications is discussed.