We examine the effects of the interplanetary magnetic field (IMF) orientation and solar wind dynamic pressure on the solar wind proton precipitation to the surface of Mercury using a hybrid-kinetic model. We use our model to explain observations of Mercury's neutral sodium exosphere and compare our results with MESSENGER observations. For the typical solar wind dynamic pressure at Mercury our model shows a high proton flux precipitates through the magnetospheric cusps to the high latitudes on both hemispheres on the dayside, centered near the noon meridian with ∼11° latitudinal extent in the north and ∼21° latitudinal extent in the south, which is consistent with MESSENGER observations. We show that this two-peak pattern is controlled by the radial component (Bx) of the IMF and not the Bz. Our model suggests that the southward IMF and its associated magnetic reconnection do not play a major role in controlling plasma precipitation to the surface of Mercury through the cusps. We found that the total precipitation rate through both of the cusps remain constant and independent of the IMF orientation. We also show that the solar wind proton incidence rate to the entire surface of Mercury is higher when the IMF has a northward component and nearly half of the incidence flux impacts the low latitudes on the nightside. During extreme solar events (e.g., coronal mass ejections), our model suggests that over 70 nPa solar wind dynamic pressure is required for the entire surface of Mercury to be exposed to the solar wind plasma.