Origin of the term "fat chance"
Solution 1:
I believe this simply derives from one sense of the word fat:
fat 2a well filled out : of sizable proportions : THICK <a ~ letter> <a ~ volume of verse> : BIG <a resistor spark plug ... permits a wider gap, thus a fatter hotter spark — Newsweek> : unusually large <he had to pay a ~ price to move his factory — Martin Turnell> [Websters 3rd New Int'l Dictionary]
The term is used ironically. At face value it means there is a large chance of something happening, but underneath it really means there is a slim chance after all.