The kinetic approach is applied to calculate oscillations of a surface plasmon linewidth in a spheroidal metal nanoparticle embedded in any dielectric media. The principal attention is focused on the case, when the free electron path is much greater than the particle size. The linewidth of the plasmon resonance as a function of the particle radius, shape, dielectric constant of the surrounding medium, and the light frequency is studied in detail. It is found that the resonance plasmon linewidth oscillates with increasing both the particle size and the dielectric constant of surrounding medium. The main attention is paid to the electron surface-scattering contribution to the plasmon decay. All calculations the plasmon resonance linewidth are illustrated by the example of the Na nanoparticles with different radii. The results obtained in the kinetic approach are compared with the known ones from other models. The role of the radiative damping is discussed as well.