Aims: The Si i 10 827 Å line is commonly used for spectropolarimetric diagnostics of the solar atmosphere. First, we aim at quantifying the sensitivity of the Stokes profiles of this line to non-local thermodynamic equilibrium (NLTE) effects. Second, we aim at facilitating NLTE diagnostics of the Si i 10 827 Å line. To this end, we propose the use of a relatively simple silicon model atom, which allows a fast and accurate computation of Stokes profiles. The NLTE Stokes profiles calculated using this simple model atom are very similar to those obtained via the use of a very comprehensive silicon model atom. Methods: We investigate the impact of the NLTE effects on the Si i 10 827 Å line by means of multilevel radiative transfer calculations in a three-dimensional (3D) model atmosphere taken from a state-of-the-art magneto-convection simulation with small-scale dynamo action. We calculate the emergent Stokes profiles for this line at the solar disk center and for every vertical column of the 3D snapshot model, neglecting the effects of horizontal radiative transfer. Results: We find significant departures from LTE in the Si i 10 827 Å line, not only in the intensity but also in the linearly and circularly polarized profiles. At wavelengths around 0.1 Å, where most of the Stokes Q, U, and V peaks of the Si i 10 827 Å line occur, the differences between the NLTE and LTE profiles are comparable with the Stokes amplitudes themselves. The deviations from LTE increase with increasing Stokes Q, U, and V signals. Concerning the Stokes V profiles, the NLTE effects correlate with the magnetic field strength in the layers where such circular polarization signals are formed. Conclusions: The NLTE effects should be taken into account when diagnosing the emergent Stokes I profiles as well as the Stokes Q, U, and V profiles of the Si i 10 827 Å line. The sixteen-level silicon model atom proposed here, with six radiative bound-bound transitions, is suitable to account for the physics of formation of the Si i 10 827 Å line and for modeling and inverting its Stokes profiles without assuming LTE.