One time Henri Berestycki was riding the Paris subway on the way to work and doing some calculations. All of a sudden, an elderly lady sitting across from him said: "Why don't you multiply by alpha and integrate by parts?" This did not solve his problem, but it was a reasonable thing to do.
It turned out the old lady had once worked with Lebesgue. She remembered J.L. Lions as a "clever lad."
I heard this story from my advisor Klaus Kirchgaessner who had heard it from Berestycki himself.