Origin of "in a pig's eye"
Solution 1:
The entry for pig in the Oxford English Dictionary has an entry for the phrase, in a pig's eye:
colloq. (chiefly N. Amer. and Austral.). (in a) pig's eye (also ear, arse) : used as a derisive retort expressing emphatic disbelief, rejection, or denial.
The listed uses are:
1847 J. J. Oswandel Notes Mexican War (1885) iii. 163 Mr. Nicholas P. Trist‥is on his way to negotiate with the Mexican government to make peace. How are you peace—peace in a pig's eye.
1876 Oakland (Calif.) Daily Evening Tribune 17 Mar. 3/7 ‘Bought this mare for $16‥’. ‘In a pig's eye you've bought her for $16’.
1951 E. Lambert Twenty Thousand Thieves 322 ‘Pig's arse to that!’ another voice cried. ‘A jack-up—that's the shot.’
1968 W. Garner Deep, Deep Freeze ix. 110 ‘One stops short of probing the private lives of people for whom one has a regard.’ ‘In a pig's ear!’ she said vulgarly. ‘If duty called you'd have a man under the bed on my honeymoon.’
1992 O. S. Card Lost Boys (1993) vi. 154 ‘She must not have any idea of the effect of her words then’‥. ‘In a pig's eye.’
So the first recorded use was in 1847, and by this time the OED says that it was already being used as a "derisive retort". As the phrase is chiefly from North America and Australia, it is highly unlikely that this is Cockney rhyming slang. However, the article does say that "in a pig's arse" is an actual variant. One of the included uses (see 1951, E. Lambert) uses arse instead of eye.
Solution 2:
The entry in phrases.org.uk has more convincing information about the etymology of this phrase:
in a pig's eye - never, highly unlikely
Whether the originator of the saying meant that a poor idea was something to put in a pig's eye or that it would look bad to a pig's eye is a matter of speculation. As an expression of scorn the expression was picked up in 1872 by Petroleum V. Nasby (David Locke) in one of his satirical newspaper columns: 'A poetical cotashun.which.wuz, -- 'Kum wun, kim all, this rock shel fly From its firm base - in a pig's eye.'" From "The Dictionary of Cliches" by James Rogers (Ballantine Books, New York, 1985).
David Locke was an American, so the expression likely originated there and is not rhyming slang.
Solution 3:
It's probably not rhyming slang. The phrase exists in French and Spanish (at least) too. 'Dans l'oeil d'un cochon', and en 'el ojo de un cerdo', respectively. It may be that those languages have borrowed it from English, but unlikely. In Spanish there is a variation "En la parte inferior de un cerdo", which you can translate for yourself. I'm trying to find it in Italian for a little feature I'm writing today. http://www.italytravelandlife.com/tag/learn-italian/
Solution 4:
St.Paul Minnesota used to be called Pigs Eye.
There’s a Birdseye in Colorado, Frogeye in Maryland, Hogeye in Texas, and even a Pigeye in both Alabama and Ohio. For a long time there was a Pig’s Eye in Minnesota until a Catholic missionary by the name of Lucien Galtier arrived on the muddy banks of the Mississippi River in 1840 to establish a Christian community. The following year, in 1841, Galtier successfully renamed Pig’s Eye to its current name of Saint Paul after he began the construction of the Church of Saint Paul. Source
and from Encyclopedia Britannica
The first land claim was made in 1838 by tavern owner Pierre (“Pig’s Eye”) Parrant; he was closely followed by Abraham Perry. The settlement was known as Pig’s Eye Landing until 1841,
Seriously, I think this is a geographical reference to something remotely possible. No article, just "in Pigseye" as a statement of disbelief as in "This is this the finest weapon in the world". The retort of disbelief would be. "In Pigseye"!; meaning the gun might be the finest weapon in Pigseye, but nowhere else.