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The dated British-English colloquial phrase to box Harry means: to go without food, or have a meagre meal, in order to save money.

This phrase is perhaps from:
– the verb box, meaning: to strike with the fist;
– the male forename Harry, designating the Devil—cf., below, quotations 2 & 4.

The British author George Borrow (1803-1881) mentioned the phrase to box Harry in detail in the following from Wild Wales: Its People, Language, and Scenery (London: John Murray, 1862), an account of his solo walking tour through Wales in 1854 [volume 2, chapter 1, pages 2, 3 & 9]—George Borrow has “arrived at the hostelry of Mr. Pritchard”:

“This is no great place for meat,” said Mrs. Pritchard, “[…] I am afraid at present there is not a bit of fresh meat to be had. What we can get you for dinner I do not know, unless you are willing to make shift with bacon and eggs.”
“I’ll tell you what I’ll do,” said I, “I will have the bacon and eggs with tea and bread-and-butter, not forgetting a pint of ale—in a word, I will box Harry.”
“I suppose you are a commercial gent,” said Mrs. Pritchard. […] “When you said ‘Box Harry,’ I naturally took you to be one of the commercial gents, for when I was at Liverpool I was told that that was a word of theirs.”
“I believe the word properly belongs to them,” said I. “I am not one of them; but I learnt it from them, a great many years ago, when I was much amongst them. Those whose employers were in a small way of business, or allowed them insufficient salaries, frequently used to ‘box Harry,’ that is have a beef-steak, or mutton-chop, or perhaps bacon and eggs, as I am going to have along with tea and ale instead of the regular dinner of a commercial gentleman, namely, fish, hot joint and fowl, pint of sherry, tart, ale and cheese, and bottle of old port, at the end of all.”
[…]
Having despatched my bacon and eggs, tea and ale, I fell into deep meditation. My mind reverted to a long past period of my life, when I was to a certain extent mixed up with commercial travellers, and had plenty of opportunities of observing their habits, and the terms employed by them in conversation […]—and then I remembered how often I had bothered my head in vain to account for the origin of the term “box Harry,” and how often I had in vain applied both to those who did box and to those who did not “box Harry,” for a clear and satisfactory elucidation of the expression—and at last found myself again bothering my head as of old in a vain attempt to account for the origin of the term “boxing Harry.”

These are, in chronological order, the earliest occurrences of the phrase to box Harry that I have found:

1-: From a letter, dated London, Tuesday 23^rd March 1802, that the British mechanical engineer and inventor Andrew Vivian (1759-1842) wrote to his cousin, the British mechanical engineer and inventor Richard Trevithick (1771-1833)—letter published in Life of Richard Trevithick, with an account of his inventions (London: E. and F. N. Spon, 1872), by Francis Trevithick (1812-1877) [volume 1, chapter 7, page 116]:

I arrived in town yesterday about three o’clock. I boxed Harry (that is, I went without dinner) that no time might be lost.

2-: From Epitaph on a Traveller, published in The Tyne Mercury; Or, Northumberland and Durham and Cumberland Gazette (Newcastle-upon-Tyne, Northumberland, England) of Tuesday 29^th June 1802 [page 4, column 5]:
—Note: The line “who knows but Harry may now be boxing him?” seems to refer to the Devil:

Here resteth the body of
T***** B*****,
late of Manchester,
who died on a journey through Scotland,
May 3, 1798, aged 30.
[…]
the fact is—he died poor:
[…]
[…] his last journey being over, there is now
no riding double stages to make up lost time:
nor boxing Harry
to make up his cash account!
who knows but Harry may now be boxing him?
The final balance
of the good and evil actions of his life is now stricken!
and here he rests in hope,
that it may be found to his credit
on the judgment day,
in the grand ledger of everlasting happiness.

3-: From one of the unconnected paragraphs making up a column of miscellaneous news-items, published in the Chester Chronicle. Cheshire & North Wales General Advertiser (Chester, Cheshire, England) of Friday 30^th June 1815 [page 3, column 6]—here, the meaning of the phrase to box Harry is obscure:

Lord and Lady Combermere are at Cheltenham.
Carter and the Chicken have been “boxing Harry,” at Northwich.”
The Staffordshire Agricultural Society meet on the 4^th of July.

4-: From an account of a trial that took place at Gloucester Assizes on Tuesday 6^th April 1819, published in The New Times (London, England) of Monday 12^th April 1819 [page 3, column 4]—the phrase to raise the wind means to procure money or the necessary means for some end:
—Note: In this text, William Burton explicitly mentioned the Devil:

William Burton, age 33, stood capitally indicted for the wilful murder of Wm. Syms, in the month of November last.
[…] The prisoner, shortly antecedent to the disappearance of Syms, had been repeatedly applying to Hurd to lend him a few shillings. His shoes had accidentally burst, and he then declared “that he was unable to refit himself, and that he must box Harry to raise the wind to buy himself a pair of shoes, though he should go to the devil for it.”

5-: From Bear-Baiting in the Days of Elizabeth, and Boxing in our own, published in Bell’s Life in London, and Sporting Chronicle (London, England) of Sunday 24^th November 1822 [page 307, column 2]—here, the meaning of the phrase to box Harry is obscure:

The Magistrates, by every proper means, prevent, as much as possible, the exhibitions of the prize ring, so that the “peace,” is never broken with their acquiescence. But whatever may be said of men who voluntarily box, to prevent their boxing Harry, it is our conviction that were it put down altogether, a half century would prove the baneful effects of the enactment by the introduction of some more dreadful means of decision.

6-: From The Stage Coach, a short story published in The Scots Magazine (Edinburgh, Midlothian, Scotland) of September 1824 [page 351, column 2]:

“Well,” said Mr White, “have you been boxing Harry?” “No,” said I, “I have had my breakfast in the kitchen […]; but the consequence will be, that I shall have to dine with Duke Humphrey, for my finances are getting so low, that I shall soon have pockets to let at a low rent.”