In British English, chiefly in the phrase to make a pig’s ear (out) of something, the noun pig’s ear is colloquially used to designate a mess, a botched job.

According to Dictionary of English Phrases (Penguin Books, 2008), by Robert Allen, this colloquial use refers to the fact that the pig’s ear is regarded as the least valuable part of the animal when slaughtered—cf. the proverb you can’t make a silk purse out of a sow’s ear, meaning: you can’t create a fine product from inferior materials.

However, according to A Dictionary of Slang and Unconventional English (Routledge & Kegan Paul, 1984), by Eric Partridge and Paul Beale, pig’s ear is probably a euphemism for pig’s arse. This hypothesis is supported by the use of the noun pig’s arse to designate a mess, a botched job.

The following are, in chronological order, the earliest occurrences that I have found of this colloquial use of:
1-: the noun pig’s ear,
2-: the noun pig’s arse:

 

1-: pig’s ear

 

1.1-: From At Last—The HON Man Brings Home The Bacon!, published in The Sunday Post (Glasgow, Lanarkshire, Scotland) of Sunday 21^st June 1964 [No. 3,069, page 8, column 3]:

“Catch his tail,” cried Gordon. “It won’t hurt him.”
I did. He stopped dead in his tracks.
But when I upended him to put him between my knees, oh dear. Master pig howled blue murder.
I stuck a finger in his mouth to hold back his tongue.
He promptly snapped his mouth shut.
Yiaow! Let go, you little brute.
I’d to prise his jaws apart to get my finger out.
“Aye, you made a pig’s ear of that,” said Gordon.

1.2-: From ‘Reduce school holidays’ plea, published in the Manchester Evening News and Chronicle (Manchester, Lancashire, England) of Tuesday 29^th September 1964 [No. 29, 604, page 5, column 7]:

STOCKPORT mothers, working round the clock for two weeks, have collected over 1,000 signatures for a petition protesting about the new nine-week summer holidays for their children.
[…]
Mrs Betty Fairclough, of Reddish Road, Stockport, said: “I think they made a pig’s ear of it. We couldn’t plan to go away. And half the shops are open.”

1.3-: From Tremendous chances for young designers, published in The Guardian Journal (Nottingham, Nottinghamshire, England) of Saturday 2^nd January 1965 [No. J43,322 G33,842, page 5, column 7]:

A comment on the subject of design by Mr. D. P. Bromley, Director of the Midlands’ Building and Design Centre: “When you are sitting in a traffic jam at Canning Circus, you think ‘someone designed this—and made a pig’s ear of it.’”

1.4-: From the column Amateur Stage, by David Mackenzie, published in the Rochdale Observer (Rochdale, Lancashire, England) of Wednesday 21^st April 1965 [No. 9,766, page 5, column 3]:

Amateurs now know that shoddy workmanship and production will be met with audience contempt and consequent drop in patronage. “Acting unpaid,” in military parlance, means holding the unpaid rank for a trial period; if you do well you get the “stripe” (paid)—if you make a pig’s ear of it they tear you off a “strip.”

1.5-: From Vickers fights back, by Anthony Bambridge, business editor, published in The Observer (London, England) of Sunday 27^th November 1966 [page 6, column 2]:

Since the war plenty of companies have made a pig’s ear of steel, even more have come unstuck in shipbuilding, none have had a smooth ride in aircraft. But only Vickers has suffered at the hands of all three.

1.6-: From the column One woman’s view, by Vivien Tomlinson, published in the Liverpool Daily Post (Liverpool, Lancashire, England) of Tuesday 11^th April 1967 [No. 34,769, page 6, column 7]—the Labour politician Kenneth Robinson (1911-1996) was the Minister of Health from 1964 to 1968:

Mr Kenneth Robinson is in the news again. He has now made the perfect pig’s ear of an official statement: no one seems sure whether to take his words on the Pill as either reassuring or a warning.

 

2-: pig’s arse

 

2.1-: From Grouching at the Groucho, by Peggy Vance, published in The Bookseller (London, England) of Friday 25^th September 1992 [No. 4,527, page 913, column 2]:

Candidate 1: Ya, I mean, you know, it’s just, like such a drag, I mean, I wouldn’t go, but, like, there’s just, like, serious pressure, you know? I’m kind of a straight talker, and they’re, like, relying on me, begging me to, like, reel in the deals. It’s a bit of a pig’s arse, but, like, there you have it; got to take the “book” by the horns, ya, and just, like, brazen it out. Harsh but true, ya?

2.2-: From the Daily Mirror (London, England) of Monday 21^st June 1993 [page 36, column 3]—Graham Taylor (1944-2017) was the manager of the England national football team from 1990 to 1993:

TAYLOR’S brutal verdict on England’s trail of disaster
‘I’ve made a real pig’s a*!@ of it’
He’s not scared of the bullet
GRAHAM TAYLOR admitted he made “a pig’s arse” of managing England in Norway.
But he revealed that he can “live with the sack” should he fail again in the World Cup ties against Poland and Holland.
[…]
He goes home to his wife Rita and to his dogs, pleading to be left alone. After his ill-advised “headless chickens” remark in Poland, and his “pig’s arse” in Norway, Taylor is literally going to the dogs. Some might say he is already there.

2.3-: From a letter to the Editor, published in the Daily Mirror (London, England) of Thursday 24^th June 1993 [page 37, column 5]:

Ear, ear, sir
GRAHAM Taylor now says England made a “pig’s arse” of their performance against Norway.
Someone should tell him the acceptable phrase for a poor show is “pig’s ear”. We have enough to put up with without expletives from someone who should know better.