The renormalization of the electron g factor by the confining potential in semiconductor nanostructures is considered. A new effective k . p Hamiltonian for the electronic states in III-V semiconductor nanostructures in the presence of an external magnetic field is introduced. The mesoscopic spin-orbit (Rashba type) and Zeeman interactions are taken into account on an equal footing. It is then solved analytically for the electron effective g factor in symmetric quantum wells (g(QW)*). Comparison with different spin quantum beat measurements in GaAs and InGaAs structures demonstrates the accuracy and utility of the theory. The quantum size effects in g(QW)* are easily understood and its anisotropy Lambda g(QW)* (i.e., the difference between the in-plane and perpendicular configurations) is shown to be given by a mesoscopic spin-orbit effect having the same origin as the Rashba one.