In a pig's eye
An expression of emphatic disbelief.
When compared the human's eyes, pig's eyes are relatively small and the expression 'a pig's eye' has been used to denote small eyes since the 17th century. The British poet Richard Flecknoe put that usage into print in 1658, in Enigmaticall Characters:
She have the spirit in her of twenty school-mistresses, looking with her Pigs-eyes so narrowly to her charge.
The phrase 'in a pig's eye' emerged in the USA in the 19th century and, while it is used in Australia, it hasn't travelled to the UK and its meaning is generally unknown there. The expression's use to indicate incredulity could be related to the earlier 'pig's eye' meaning but, if so, it isn't clear how. It is in the same linguistic area as 'pigs might fly' - so it might also be related to that.
Enough of speculation. We do know that 'in a pig's ear' was a variant of 'in a pig's eye' and that the first known example of the phrase in print is in Jacob Oswandel's Notes on the Mexican War 1846-1848. Oswandel was a volunteer in Pennsylvania's First Regiment. He seems fond of the 'in a pig's eye' expression and used it five times in his account of the war, for example:
The Publicanos de Mexicanos were all anxious to see the new arrivals, they having been informed that our regiment was a whole division of about eight thousand men (in a pig's eye).
Confusingly, the pig's ear phrase also has more than one meaning; three to be precise. See: the meaning and origin of 'pig's ear' for more on that.