In British English, the term (white) rabbit(s) is an incantation to be muttered to oneself when waking up on the first day of the month in order to secure good luck for the whole of that month.

The reason that those specific words were chosen is unknown.

The earliest mention of this incantation that I have found is from the following, by William Gerish, a local antiquarian, which appeared in Norfolk and Norwich “Notes and Queries”, published in The Norfolk Chronicle and Norwich Gazette (Norwich, Norfolk, England) of Saturday 6^th December 1902:

Norfolk Folk Lore.—(Note 481 Continued).
Superstitions Connected with the Animal World.

[…]
Kittens.
Kittens born in March are said to be dirty in their ways, and so some people object to rear them as household pets.
Rabbit.
If on waking on the first morning of a month you repeat the word “Rabbit” three times before speaking to anyone you will have a present before the month is out.
Hare.
It is unlucky for a hare to cross your path.

The second-earliest mention of the incantation that I have found is from an article about New Year’s resolutions, by ‘L. W. B.’, published in the Thanet Advertiser and Pullen’s Kent Argus (Ramsgate, Kent, England) of Friday 6^th January 1928—the manner in which the author evokes the incantation indicates that it was well established at that time:

Of course, there are such beings, poor souls, who have no vices. Theirs is a sad lot. The New Year means nothing to them. At least, nothing more than a new calendar and the necessity of saying “White rabbits” on the first of the month for luck. For them is not that glorious feeling of cleanliness and righteousness that comes to the sinner who, on January One, steps out of bed at the first tinkle of the alarm, lights the fire, takes up a cup of tea to his spouse, makes his own breakfast, and goes whistling to work three minutes before time.
I once knew one of these model young men. He was much perturbed because of his spotless past, and because the New Year held no joy for him as it did for his more sinful (or honest) brethren. At last he hit upon a happy idea, and on December the thirty-first he set out to gather together such a crop of vices that he made Captain Blood look like an angel. He smoked, he drank, he gambled. He swore at the cat and put pepper on the dog’s dinner. He kissed a girl—twice, and even put his fingers to his nose as he passed the local council chamber. In one short day he became a perfect bad lad, the doggiest of the dogs, and, said he: “To-morrow I will put aside all my evil ways, I will not be as other men. I will keep a diary, and I will give up all my bad habits.”
Alas and alack! His plans, like those quoted by the poet, went agley. The liquor, to which he was not accustomed, made him sleep so soundly that he overlaid and was late for work, thereby losing a quarter. He awoke with such a head that he swore heartily when he tripped on the bedroom mat. He forgot to say “White rabbits” before breakfast, thereby losing all his luck for the ensuing month, and, finally, he received from the girl he had so daringly kissed a letter saying that she accepted him although his proposal was somewhat unorthodox, and that her mother would not mind living with them after they were married.

This amusing item is from the column Interesting Pars, published in the Kensington Post and West London Star (London, England) of Friday 6^th December 1929:

RABBITS!

Rabbits have been much in the news of late. A motorist has just put on record the fact that he had to travel at thirty-four miles an hour to keep up with a rabbit that was running in front of his car. Then, Russia has decided to raise rabbits by the million for food.
But the rabbit, in spite of its smallness and timidity, has had a way of making a noise in the world. It is a curse to gardeners and farmers even in this country, and in Australia it quickly became a national menace.
In this connection a good story is told of a new-comer to the Island Continent—a Scot—who asked if there were many Scots in Australia.
“Yes,” was the immediate reply, “but our real plague is the rabbits!”
But—even though the Australians won’t believe it—rabbits are luck-bringers. Say “White Rabbits” three times first thing on the first day of each month, and you’re in for four weeks of good fortune! So, at least, runs the story.

The following is from Readers’ Letters, published in The Devon and Exeter Daily Gazette (Exeter, Devon, England) of Tuesday 8^th December 1931:

“RABBITS.”

To the Editor of the Daily Gazette.
Sir,—A few days ago I heard an interesting bit of folk-lore:—If you desire to receive a present before the month is out, repeat the words “White rabbit” three times on the morning of the first of the month before speaking to anyone. Can any reader of the “Gazette” tell me the origin of this superstition?
In my native county of Somerset it is thoroughly believed that if a person says “Rabbits” or “White rabbits” when awakening on the first day of the month a present will follow. In the North of England the last word to be uttered on the last day of the month has to be “Hares,” and the first word spoken the following morning must be “Rabbits,” in order to secure the present—or good luck—during the ensuing month. I believe years ago the custom was followed in the county of Norfolk, but I have never been able to ascertain the origin of the custom.
Yours truly,
MOON DOUTER.
December 7th, 1931.

The English folklorists Iona Opie (1923-2017) and Peter Opie (1918-1982) described this British custom in The Lore and Language of Schoolchildren (Oxford University Press, 1959):

‘On the first morning of the month’, notes a typical informant, ‘before speaking to anyone else, one must say “White rabbits, white rabbits, white rabbits” for luck. Subject to minor modifications the utterance of this spell appears to be the accepted routine throughout Britain. Some children feel it is enough just to cry ‘Rabbits’, as long as it is the first word they pronounce. Others, though not many, believe it is necessary to say ‘Hares’ last thing the previous night, as well as saying ‘Rabbits’ in the morning. In Romford a boy says that ‘White Rabbits’ must be intoned three times, ‘after the first foot has touched the floor when getting out of bed, and not after the second foot has touched the floor or it will bring bad luck’. In Liverpool the first of the month is known as ‘Bunny Rabbit Day’. In Luncarty, near Perth, it is considered especially lucky if ‘Rabbits’ is cried on the first of May. Others in Perthsire hold that it is important to say ‘Rabbits’ when there is an R in the month. Radnorshire children, or some of them, assert that ‘Black rabbit’ should be shouted on the eve of the new month, and ‘White rabbit’ shouted in the morning; and the same view appears to be held in parts of Devon, for a South Molton girl warns that while it is lucky to say ‘White rabbit’, ‘if you say black rabbit on the first day of the month you have bad luck all the month’.

In Superstitions Die Hard, published in The Sphere (London, England) of Saturday 27^th May 1961, Brian Vesey-Fitzgerald described the manner in which the incantation should be said:

Most people, I think, have at least one [superstition], though few will admit it. Stand in a busy street and notice the number of people who will step off the pavement, despite the traffic, rather than walk under a ladder. Maybe you ’re one? Think of all the people who “touch wood.” Maybe you ’re one? All the people who throw salt over the left shoulder should they happen to spill some at table: all the people who cannot abide crossed knives on the table: all the people who say “white rabbits” on the first of the month.
But that is skimping it. If you really want a lucky month, then you should come downstairs backwards, repeating “white rabbits” on each step. I don’t know anybody who does this now—but perhaps you do—but I did once catch my sister’s nurse, a stout body with a moustache and a formidable temper, in the act. They were steep stairs, and I stood at the bottom and watched her with some very un-Christian thoughts in my mind.