The effect of the main surfactant properties such as, solubility, surface activity, and diffusion coefficient on the behavior of a system with a surfactant droplet under the free water surface is studied numerically for a semi-infinite model system. The system reveals instability which is manifested by auto-oscillation of the surface tension. The induction time before the beginning of auto-oscillations and amplitude and period of the oscillation are dependent on the surfactant properties. Increase of the above parameters leads to a decrease of the induction time but the degree of their effect is different. The amplitude of the oscillations is mainly dependent on the surfactant solubility.