Of American-English origin, the colloquial, sarcastic phrase what has that got to do with the price of ——? is a rhetorical question calling attention to a non-sequitur or irrelevant statement or suggestion made by another person.

In this phrase, the noun following the price of is irrelevant to the context in which it is used. For example, in the following Readers Write letter by one Steve Logsdon, published in the Messenger-Inquirer (Owensboro, Kentucky, USA) of Tuesday 6^th January 2015, eggs in China is irrelevant to the context in which it is used:
—Note: Steve Logsdon used what has that got to do with the price of eggs in China?, which is perhaps a conflation of what has that got to do with the price of eggs? and of what has that got to do with the price of tea in China?:

In response to a Readers Write letter published in the Dec. 31 M-I written by James Goodall titled “Don’t forget that law enforcement keeps us safe every day,” I can only say that, having over the years read more than a few of his jingoistic profferings, I expected no less illogical and simplistic innuendo from him.
[…]
Mr. Goodall’s Readers Write letters all seem to invariably allude to military service, as in his mentioning that “A good many of our law enforcement officers are former military personnel who put their lives on the line for our country.”
What has that got to do with the price of eggs in China? I, too, am an honorably discharged USAF veteran, but I don’t seize every opportunity to tout it.

The earliest occurrences that I have found of the phrase what has that got to do with the price of ——? and variants are as follows, in chronological order:
—Note: In a few of the quotations below, the context does not clearly indicate whether the question was rhetorical or was eliciting an answer—cf., for example, in quotation 7, the remark “who had mistaken the object of the meeting”, which (if not ironical) would indicate that the question was eliciting an answer:

1-: From Too Fast and Too Slow: Or, Chance and Calculation, by the U.S. author James Kirke Paulding (1778-1860), published in The New-York Mirror: A Weekly Journal, devoted to Literature and the Fine Arts (New York City, New York, USA) of Saturday 10^th March 1832:

Without stopping to let himself be eaten up with chagrin at these provoking disqualifications which beset him at every new trial, Frank forthwith placed himself in the counting-house of an eminent merchant, who had grown rich by the mere force of the instinct of saving. One day the old gentleman was directing him to make an entry in his day-book of some sale or other.
“Is it according to reason and experience?” asked Frank, who was determined to be right this time. “Does it correspond with the seven sciences?”
“The seven devils,” exclaimed the merchant, whom gout and money had made as testy as a young belle of a rainy day. “The seven sciences, and reason and experience! What the plague have these to do with the price of tobacco? I can tell you what, sir, no more of such nonsense, or you won’t do for a counting-house.”
“I think so too,” said Frank, and sliding from his high three-legged stool, quietly put on his hat, and was walking home, meditating on what he should turn his attention to next.

2-: From The Political Destination of America, and the Signs of the Times, by the U.S. Unitarian minister, transcendentalist and social reformer Theodore Parker (1810-1860), published in The Massachusetts Quarterly Review (Boston: Coolidge & Wiley) of December 1848:

The antiquary unrolls his codex, hid for eighteen hundred years in the ashes of Herculaneum, deciphers its fossil wisdom, telling us what great men thought in the bay of Naples, and two thousand years ago. “What do you tell of that for?” is the answer to his learning. “What has Pythagoras to do with the price of cotton? You may be a very learned man; you can read the hieroglyphics of Egypt, I dare say, and know so much about the Pharaohs, it is a pity you had not lived in their time, when you might have been good for something; but you are too old-fashioned for our business, and may return to your dust.”

3-: From The Westchester Herald (Ossining, New York, USA) of Wednesday 29^th August 1855:

We just make a note here, that Adam Bird, E. H. Wildey, and B. Gage Berry, with some others of whose cases we are not advised, have been deemed, upon second thought of the whig powers that be, unfit or incapable of filling the posts to which they had been appointed through their means. But what has this to do with the price of Manilla rope?

4-: From the answers to the queries received by the Editors, published in Pomeroy’s Democrat (New York City, New York, USA) of Wednesday 26^th April 1871:

Here is another letter:
Will you please tell me, through Pomeroy’s Democrat, how many of the Kings of England were Masons:
We would answer the above question with great pleasure, but, really, it makes no difference to us whether any of them were or not. What has that to do with the price of a pair of boots?

5-: From the account of a speech delivered by Edwin H. Coates, of Philadelphia, during a Republican rally held at City Hall, published in The Daily Post (Pittsburgh, Pennsylvania, USA) of Friday 22^nd September 1871:

I have raised a large family (a voice—good enough)—and all have earned their own bread—my son went out to fight the rebs, and when he came back he met me in the bath house (the Great Unwashed)—and said, “father, I come home too soon—there are men living down South who ought to be under the ground (very benevolent)—I don’t understand your local politics.”
“And he who allows oppression justifies the crime.”
A gentleman in a front seat—What has that got to do with the price of oats?

6-: From the account of a speech delivered by the Irish-born U.S. orator John Pope Hodnett (c.1847-1913) during a meeting of workingmen of the District, held at Lincoln Hall, published in the National Republican (Washington, District of Columbia, USA) of Thursday 4^th June 1874:

The whole District government ought to be sent to the Dry Tortugus or happy land of Canaan. I don’t care which, only its [sic] where I can’t meet them again. [Immense applause, while a voice cried, “Happy land of Canaan is good; Popp, you’re a smooth pressed-brick.”] Popp bowed in recognition and continued, Napoleon planned a scheme once to conquer all the world and a portion of New Jersey for the purpose of making one grand universal republic. The immortal, spotless, pure, virtuous, high-minded, big-souled, long-legged, straightforward, go-ahead, brilliant Daniel O’Connell was of the same idea as Napoleon, and so am I, ladies and gentlemen, and he smote his breast. [“What’s that got to do with the price of bread,” vociferated a sonorous voice in a back seat.] Johnny Popp said he would show the similarity presently, and continued.

7-: From the account of a speech delivered by a politician called A. M. Green during a public meeting held in the Legislative Hall—account published in the National Republican (Washington, District of Columbia, USA) of Thursday 16^th July 1874:

Things became serious, and Mr. Green, planting his knuckles on the American flag, which shrouded the speaker’s desk, cried, “War! war! I’m for a war of extermination, to be kept up until the old shebang has been cleanly swept and revolutionized; until every man in office is thrown out, and everyone out is thrown in; and then, and not until then, will we have a government for the people, by the people and”—[“What has all this got to do with the price of provisions?” interrupted a day laborer, who had mistaken the object of the meeting.]

8-: From the account of speeches delivered by the Independents on the Public Square, published in the Republican Banner (Nashville, Tennessee, USA) of Sunday 2^nd August 1874:

Robertson was hissed, hooted and groaned at, and McGowan shared the same fate. His words were perfectly unintelligible, being drowned in the groans, yells and exclamations uttered every moment. He mumbled out a few words amidst cries of “Take him down,” “D—n the nigger, down with him,” “Put a tin roof on him,” “Chuck him in the gutter,” “If he can’t speak, let somebody get there who can,” “Oh, my McGowan, you’re a good un,” “Drunk and disorderly;” “Pass him back;” “Fifty dollars and costs;” “What’s that speech got to do with the price of bacon,” and similar exclamations.

9-: From one of the unconnected paragraphs making up the column Brevities, published in The Idaho Avalanche (Silver City, Idaho, USA) of Saturday 3^rd February 1877:

The sausage season will soon be round. This has nothing to do with the price of bark.

10-: From the Montgomery Daily Advertiser (Montgomery, Alabama, USA) of Wednesday 13^th February 1878:

The Bible says there shall be wars and rumors of war, and that is what’s the matter now. When Queen Vic. tackles old Aleck then the little Greeks had as well quit their blowing. But what’s that got to do with the price of cotton and shot guns on this side of the pond.

11-: From All Sorts. Curious Paragraphs Drifting on the Current of the Journalistic Sea, published in The Harrisburg Daily Patriot (Harrisburg, Pennsylvania, USA) of Friday 21^st May 1880:

The Yonkers Gazette assures a correspondent that the “Lay of the Last Minstrel” had nothing whatever to do with the price of eggs. It wasn’t that kind of a lay.

12-: From the following unsigned story, published in The Chicago Daily News (Chicago, Illinois, USA) of Tuesday 14^th February 1888:

HOW IT CAME ABOUT.

His name was Harry Sutherland. He was a young telegraph operator employed in a large office in a great city. Being of an ardent temperament he was easily impressed by the winning ways of the many young and fascinating girls who worked in the building. […]
[…] There was one modest little maiden, with blue eyes and with corn-colored hair, who watched Harry, but was too retiring to make any advances to him. […]
This modest girl was named Josie Winthrop. When six months had gone by she had treasured up a love for Harry which engrossed her whole thought. […] It happened that Harry, in company with Julia Grayson, took tea one night with Josie. […] Harry’s attention was attracted to her. He saw what a modest, retiring little maiden she was.
[…] Josie awakened a glow in his bosom which burned brighter and brighter as the days went by, until he found himself mixing her name with his telegraph messages. Many a time did a message come back to him like this:
“What has Josie got to do with the price of pork?”

13-: From The Wheeling Daily Intelligencer (Wheeling, West Virginia, USA) of Friday 27^th April 1888:

Old Subscriber:—You are right, the protective tariff has nothing to do with the price of base ball umpires.

14-: From The Morning News (Savannah, Georgia, USA) of Wednesday 30^th May 1888:

The Bustle as an Arm Rest.
From the Leary (Ga.) Courier.

It is given out that Mrs. Cleveland has discarded the bustle. But what has this got to do with the price of provisions? We hope our lady friends will not follow Mrs. Cleveland’s example in this particular. The bustle is the best arm rest ever invented, and its usefulness should not be sacrificed because one woman says it must go. Who is Mrs. Cleveland, anyhow?

The question what has that got to do with the price of tea in China? perhaps alludes to the phrase not for all the tea in China, which is first recorded in the late 19^th century and means not in any circumstances.

The earliest occurrence that I have found of what has that got to do with the price of tea in China? is from the column Nothing’s Bar’d from the Mill of R. Klare Bard, published in the Altoona Tribune (Altoona, Pennsylvania, USA) of Wednesday 30^th April 1930:

Ray ‘N’ Don have a couple of questions to ask. If I slip up, you can take your best hold and call the mistakes to my attention.
The first question is: “Does a steeplechase rider rate a trophy when he makes a dive into the water as his horse leaps a hedge or hurdle?”
Answer: The trophy would probably be a cup for the fancy diving championship.
The second question: “If the big rubber men keep stretching things, how many mucilage men will it take to make the Union stick together?”
Answer: On third Fridays in July. But what has that to do with the price of tea in China?