Originally and chiefly used with reference to hair loss treatment, the phrase to grow hair on, or upon, a billiard ball means to achieve the impossible.

The earliest occurrence of the phrase that I have found is from No Hope for the Bald Heads. But Some Comfort for Those who are Growing Bald—A Useful Preparation, published in The Sun (New York City, N.Y.) of Monday 26^th September 1881; in this article, “a physician” told the following about the soap “best known as the German green soap”:

“The soap has a disagreeable fishy odor, and the oil of lavender is added to cover it up. The preparation thus compounded has a rich orange or wine color, and a pleasant odor to which the most fastidious will hardly object. Now, I don’t mean to say that it is going to grow hair on a billiard ball. Where alopecia has lasted so long that the hair bulbs have become atrophied nothing will restore the hair on these spots, but we can save what remains.”

(Several U.S. newspapers reprinted this article in September and October 1881.)

The second-earliest occurrence of the phrase that I have found is from the description of the man who is growing bald, in The Masculine Bald Head, published in the Springfield Globe-Republic (Springfield, Ohio) of Sunday 11^th January 1885:

He precipitates himself upon the drugstore and buys more than a million consecutive washes—wash after wash—that have been patented and certified (sworn affidavits in the newspapers, with name and residence) to grow hair on a healthy billiard-ball.

The earliest British use of the phrase that I have found is from Chapter IX of Una Montgomery, a novel by the Welsh author Evan Rowland Jones (1840-1920)—as published in The Cardiff Times and South Wales Weekly News (Cardiff, Glamorgan, Wales) of Saturday 20^th October 1888:

Dr Bromley called in his assistant, and told him, laughing immoderately as he spoke, that “Lovedale prescribed whisky and potass [= potash] to cure pains in Montgomery’s lost arm. Why,” he continued, “he’ll be giving it for intra-cranial tumours next.”
“Yes,” answered the assistant, who seemed to enjoy the joke; “or to grow hair upon a billiard ball.”

The phrase has occasionally been used in advertisements for hair loss remedies—here are two examples:

1: This advertisement was published in The Philadelphia Inquirer (Philadelphia, Pennsylvania) of Wednesday 25^th February 1891:

“Oakaline” won’t grow hair on a billiard ball, but it’ll save your head if it’s going that way. 25c. a bottle.

2: This advertisement was published in Frostburg Mining Journal (Frostburg, Maryland) of Saturday 21^st September 1895:

Well,
IT IS SIMPLY
A Matter of Fact

Which has been practically demonstrated right here in Frostburg that we manufacture a

Hair Tonic

That will absolutely PREVENT Baldness!
It will not grow

Hair on a Billiard Ball,

but if there is any life or vigor left in your hair, this Tonic will stimulate and restore it.
We call it simply

Pearce’s Hair Tonic.
PRICE, 50 CENTS PER BOTTLE.
G. E. PEARCE DRUG CO.,

Main Street.

Sept 21.

A use of the phrase in an extended acceptation is attributed to the Iowa poet Paul Engle (1908-1991) in the homage paid to him by The Gazette (Cedar Rapids, Iowa) of Sunday 24^th March 1991:

There are now some 200 college writing programs, most patterned after Engle’s then-radical notion back in the mid-1930s that creative writing was a legitimate course of study.
Rather than being taught, Engle believed that writing could be nurtured. “You can’t grow hair on a billiard ball,” he liked to say, noting talent can only be developed and not created.

Douglas Robarchek (1943-2018) used the phrase humorously in his column OutFront, published in The Charlotte Observer (Charlotte, North Carolina):

– of Thursday 5^th December 1996:

Our pal Heppel—poor guy—is too gullible. He was losing his hair, and a guy said he had some stuff that would grow hair on a billiard ball, and double your money back if it didn’t work.
Poor Hep. Now, not only is he losing his hair, he has to shave his billiard balls.

– of Thursday 30^th October 2003:

Buy! Buy! Buy!
FOR SALE: OutFront Miracle Elixir.
OutFront Miracle Elixir removes warts, cleans teeth, kills tapeworms, melts away pounds without dieting, cleans drains (it will keep both you and your sink running), seals leaky automobile radiators, removes stubborn laundry stains and will remove a coat from your tongue and a wallet from your coat.
If your breath is bad, this stuff will stop it completely, and best of all, if you’re balding, it’s a powerful restorative that’s so strong it will grow hair on a billiard ball.
ALSO FOR SALE: Large selection of hairy billiard balls.
Order by midnight tonight and we’ll include an autographed nude photograph of Marie Antoinette’s mother, Mrs Gladys Antoinette; a bobble-head doll of William Howard Taft, and a half-off coupon for day-old sushi.