The best one I can think of would be: The Attempts to Resolve The Continuum Hypothesis, and The Souslin Hypothesis.
The list of tools developed to deal with the new and subtle problems that cropped up while trying to either affirm or refute these two classical problems is almost unending. The main take away from all of this work was that the universe is much more flexible than we had ever dreamed.
A short list:
- Forcing ( wiki )
- Model Theory ( wiki model theory )
- Inner Model Theory ( inner model theory )
- The Various Large Cardinal Hypothesis which have been developed ( cantors attic ) and Kunen's delimiting result ( Kunen's inconsistency theorem )