  We study analytically and numerically the renormalization of the $g$-factor
in semiconducting Rashba nanowires (NWs), consisting of a normal and
superconducting section. If the potential barrier between the sections is high,
a quantum dot (QD) is formed in the normal section. For harmonic (hard-wall)
confinement, the effective $g$-factor of all QD levels is suppressed
exponentially (power-law) in the product of the spin-orbit interaction (SOI)
wavevector and the QD length. If the barrier between the two sections is
removed, the $g$-factor of the emerging Andreev bound states is suppressed less
strongly. In the strong SOI regime and if the chemical potential is tuned to
the SOI energy in both sections, the $g$-factor saturates to a universal
constant. Remarkably, the effective $g$-factor shows a pronounced peak at the
SOI energy as function of the chemical potentials. In addition, if the SOI is
uniform, the $g$-factor renormalization as a function of the chemical potential
is given by a universal dependence which is independent of the QD size. This
prediction provides a powerful tool to determine experimentally whether the SOI
in the whole NW is uniform and, moreover, gives direct access to the SOI
strengths of the NW via $g$-factor measurements. In addition, it allows one to
find the optimum position of the chemical potential for bringing the NW into
the topological phase at large magnetic fields.
