How many parts does the latest-generation 737 share with the original 737?
The cross section and cabin pressure have remained the same. From what I can tell virtually every part is different. Even parts that might have been able to retain the same shape/loading have gotten improved materials and/or corrosion treatments. I am still looking to see whether any window panels could be common as the holes into which they go have remained the same.

Looking at a couple of generations shows major differences, allowing elimination of whole sections of the aircraft. Wings, tail, engines, pylons, landing gear, fuselage center sections, APU, interior and flight deck have all surely changed. 

Originally conceived as a 60 seat jet, the first production model was designed for 100 seats. 



The Max10 now goes to 230. 



The Max also had major improvements to the the cockpit. Even the throttle levers changed. Here is a photo comparison to the original -100 flight deck.



b737-100



b737-max

NG changes, both structural and cosmetic.




larger wing with new flaps slats
larger tail with improved rudder and elevator
simplified higher capacity landing gear
engines
new AC electrical system
new corrosion resistant materials and treatments throughout the plane
new interior seats, panels, bins, coverings
improved avionics