Resonant hyper-Raman-scattering spectroscopy was applied for studies of the exciton--LO-phonon interaction in spherical CuCl nanocrystals of different sizes embedded in a glass matrix. Both the LO and 2LO phonon bands have shown a prominent resonance with the lowest energy $1S$ confined exciton state. The ratio of the integral intensities of the LO and 2LO bands \ensuremath{\rho} was found to increase with increase of the incident photon energy, or decrease of the nanocrystal radius. The Huang-Rhys factor $S$ has been calculated as a function of the nanocrystal size from the experimentally measured values of \ensuremath{\rho} on the offset harmonic-oscillator model of electron-vibrational coupling. It has been found that $S$ increases monotonically from 0.22 to 0.7, with the nanocrystal radius decreasing from 3.6 to 1.6 nm.