Within the semiclassical approach, we calculate contributions of the Berry phase and of the Zeeman coupling of an electron moment with the magnetic field to the phase of the Shubnikov--de Haas oscillations for the surface electrons in the Bi${}_{2}$$X$${}_{3}$ family of three-dimensional topological insulators ($X$ stands for Te or Se). We also discuss a relation of the obtained results with published experimental data on the Shubnikov--de Haas oscillations for this family of topological insulators.