Hit upon
Definition 1 of 2
Pro Tip 1/3Sudden Realization
Used when an idea or solution occurs to someone unexpectedly, often after effort. 
I was puzzled for weeks, but then I hit upon the answer while showering.
Pro Tip 2/3Creative Insight
Implies a creative or original thought that seems to appear out of nowhere. 
The architect hit upon a unique design that completely changed the project.
Pro Tip 3/3Eureka Moment
Can emphasize a moment of clarity, like the sudden 'Eureka!' in discovery. 
The scientist hit upon a breakthrough while re-examining the data.