Interstellar dust plays an important role in radiative and chemical processes in galaxies. These processes depend on the dust properties such as dust abundance, grain size distribution, and chemical compositions. On the other hand, the evolution of dust is related to the interstellar properties, feedbacks, and likely the galaxy assembly history. To understand the evolution of dust properties, especially the grain size distribution, in Milky-Way-like galaxies, we apply our semi-analytical dust evolution model to the state-of-art cosmological simulation IllustrisTNG (TNG). We model the dust abundance and grain size distribution arising from stellar dust production, accretion, sputtering, coagulation, and shattering by post-processing a sample of mock galaxies from TNG. By using a cosmological simulation, we can track how the contributions of these processes respond to the evolution of gas content, metallicity and star formation rate over the lifetime of each galaxy, and also include the feedbacks. In addition, the galactic assembly history is properly taken into account by considering full merger trees of galaxies. Our predictions for grain size distributions and extinction curves therefore reflect the wide diversity of assembly histories that arise for Milky Way-like galaxies in $\Lambda$CDM. Our results broadly reproduce the Milky-Way extinction curve at $z\lesssim 1$ and the dust--metallicity relation of nearby star-forming galaxies. Furthermore, we discover that the dispersion of the grain size distribution is minimized at $z=1$. This phenomenon may be due to the evolution of the metallicity and star-formation activities in the galaxies that lead to the equilibrium of the dust evolution processes.