What part of a hurricane determines if it makes landfall?
Hurricanes make "landfall" when the center of the eye moves over land. 

From the NOAA National Hurricane Center glossary: 


  Landfall:
  The intersection of the surface center of a tropical cyclone with a coastline.Because the strongest winds in a tropical cyclone are not located precisely at the center, it is possible for a cyclone's strongest winds to be experienced over land even if landfall does not occur. Similarly, it is possible for a tropical cyclone to make landfall and have its strongest winds remain over the water. 


Below is an example image showing when landfall occurred for Hurricane Harvey: