We examine the thermocapillary-driven flow of a droplet on a nonuniformly heated patterned surface. Using a sharp-interface scheme, capable of efficiently modeling the flow over complex surfaces, we perform 2D and 3D finite element simulations for a wide range of substrate wettabilities, i.e., from hydrophilic to superhydrophobic surfaces. Our results demonstrate that the contact angle hysteresis, due to the presence of the solid structures, is responsible for the appearance of a critical thermal gradient beyond which droplet migration is possible; the latter has been reported by experimental observations. The migration velocity as well as the direction of motion strongly depend on the combined action of the net mechanical force along the contact line and the thermocapillary induced flow at the liquid-air interface. We also show that through proper control and design of the substrate wettability, contact angle hysteresis, and induced flow field it is possible to manipulate the droplet dynamics: in particular, controlling its motion along a predefined track or entrapping by a wetting defect a droplet based on its size, as well as providing appropriate conditions for enhanced mixing inside the droplet.