I have not read the book but I think it was Seveus Snape (The Half Blood Prince) who had figured out, and documented in the book, the required changes to the method to do it right. Harry was able to do it because he followed the instructions from Snape (without knowing who wrote those).
I thought he said that no student had ever produced a sufficient draft, but if he did say there was one student that it would have to be Snape because Harry was using his recipe from the Half-blood Prince potions book.
harry knew how to make it since his book belonged to the half blood prince. the half blood prince is snape. so snape is the one who made it. lily was smart but no one can beat snape in potions or dark arts
I think he only mentions that only one person has won it in the movie but doesn't mention who it is, the only known uses of Felix Felicis in the book was:
Slughorn - Used it twice; he had two perfect days in his life Harry - Used it to get the memory from Slughorn Ron, Ginny, Hermoine - They used it in the battle at Hogwart's against the Death Eaters so they could avoid curses.