“Half-and-half” is—among other combinations—half cream and half milk, a dairy product frequently added to coffee. Rules were made in 1954 by state and city health departments and the the “half-and-half” product went on sale in New York City in 1955. The “half-and-half” product has a higher fat content than milk, but a lower fat content than light cream.
 
Evidence indicates that New York City had “half-and-half” by the 1880s, when it was served with oatmeal, cracked wheat and other grains. The New York (NY) Times of January 23, 1886,  stated: “A mixture of cream and milk, known in restaurant nomenclature as half and half, is quite popular nowadays.” The earlier citations of “half-and-half” refer to something that restaurants made on the spot; the 1954-55 citations refer to a packaged product.
   
   
Wikipedia: Half and half
Half and half refers to various beverages or liquid foods made of an equal-parts mixture of two substances, including dairy products, alcoholic beverages, and soft drinks. The United States dairy product known as half and half is a mixture of one part milk to one part cream.
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Dairy product
In the United States, half and half is a very light cream typically used in coffee. Its butterfat content is about 12.5%, which makes it low-calorie and more stable in coffee. It is widely available in the United States, both in individual-serving containers and in bulk. It is also used to make ice cream. The same product is known as half cream in the United Kingdom.
 
Epicurious.com - Food Dictionary
cream
n.  Upon standing, unhomogenized milk naturally separates into two layers — a MILK FAT-rich cream on top and almost fat-free (or skimmed) milk on the bottom. Commercially, the cream is separated from the milk by centrifugal force. Almost all cream that reaches the market today has been pasteurized. (...) Half-and-half is a mixture of equal parts milk and cream, and is 10 to 12 percent milk fat. Neither half-and-half nor light cream can be whipped.
 
(Oxford English Dictionary)
half-and-half, phr.
N. Amer. A mixture of milk and cream, often used in coffee.
1890 Olean (N.Y.) Weekly Democrat 15 May 7/4 Half-and-half, a mixture of cream and milk, is often called for, and occasionally a dish of pure cream.
1915 Washington Post 4 May 4 (advt.) A hot weather luncheon—Bowl of ‘half-and-half’—half cream, half milk—a pinch of salt, and Uneeda Biscuit.
1954 N.Y. Times 13 July 25/2 In recent years dairy companies have been marketing ‘half and half’ in Chicago and other parts of the country.
1981 Bon Appétit Nov. 23/3 (advt.) How to make a Sabra Egg Cream, 2 oz. Sabra Liqueur. 2 oz. half-and-half. Pour over ice.
2002 Dallas Morning News (Electronic ed.) 23 June H1 Wouldn’t you rather have half-and-half in your coffee than skim milk?
 
23 January 1886, New York (NY) Times, “Milk Instead of Whisky,” pg. 2:
Milk can be had by the glass at most of the drug stores that have soda-water fountains, and many hundred quarts are daily sold in that way in this city and Brooklyn. A mixture of cream and milk, known in restaurant nomenclature as half and half, is quite popular nowadays. At lunch rooms, this same combination is largely used to serve oatmeal, cracked wheat and other grains in. It is this mixture of half cream and half milk that is given considerably to people suffering from lung troubles who find difficulty in retaining on their stomachs cod liver oil in any of its emulsions or other prepared forms.
   
2 March 1887, Newark (OH) Daily Advocate, pg. 2, col. 3:
Senator-elect Hiscock takes nothing but “half and half,’ with bread—that is, half milk and half cream.
   
5 June 1889, San Francisco (CA) Bulletin, pg. 4:
Half and Half.
Customer—Here, waiter. I ordered a bowl of half and half; this isn’t it.
Waiter—Yes, sah, that’s the very article.
Customer—What? Do you mean to tell me that this is half milk and half cream?
Waiter—Oh, no, sah; I mean half milk an’ half watah, sah.
—American.
 
29 January 1903, St. Albans (VT) Daily Messenger, pg. 3:
First the appetite is sharpened a bit, when a bowl of “half and half” (cream and milk) is placed before each.
     
Google Books
Industrial Medicine
Easton, PA: American Academy of Medicine Press
1915
Pg. 71:
Very frequently the writer has stepped into a lunch room for a bowl of “half-and-half” milk and cream with bread and has noticed incidentally a young wage-earner on his right ordering pie “a la mode” and coffee, while perhaps the young man on his left is regaling himself for his four or five hours of afternoon work with strawberry shortcake and coffee.
     
Google Books
Long Ever Ago
By Rupert Hughes
New York, NY: Harper & Brothers
1918
Pg. 272:
He plied her with subtle soup, with fat pork chops and fried potatoes, with more of the same, with glasses of half and half (half milk and half cream), with jellies and with comfits, and finally for a climax he set before her that last word in fatteners, apple pie with ice-cream on it.
   
9 December 1920, Oshkosh (WI) Daily Northwestern, pg. 15, col. 5:
CEREAL WITH HALF
AND HALF, 10 CENTS
(By Associated Press)
Milwaukee—Further declines in prices of dishes served at downtown restaurants are taking place, it was reported today. At one restaurant all cereals generally included in breakfast orders were cut to 5 cents. With half milk and half cream, the cereals are 10 cents a dish.
 
13 July 1954, New York (NY) Times, pg. 25:
NEW CREAM BLEND
APPROVED BY CITY
Sale of “Half-and-Half” Will
Start After Rules Are Set
The Board of Helath gave its blessing yesterday to “half and half,” a mixture of milk and cream that can be marketed below the price of cream.
 
Detailed regulations for the blend, which is designed for use with cereals and fruits and in coffee, will be submitted to the board at its next meeting Sept. 12. Meanwhile no date will be set for beginning the sale of the blend.
 
It is expected the product will have a butter-fat content of at least 10 per cent, or nearly mid-way between milk, with a minumum of 3.3 per cent, and light cream, with 18 per cent. THe reatil price of light cream ranged from 18 to 24 cents a half-pint, the Health Department said.
 
The board’s action was in line with an amendment to the Agriculture and Market Law adopted by the Legislature at the last session to permit the sale of the intermediate product.
 
Officials believe that the dairy industry here will wlecome the opportunity to sell “half and half.” They explained that even after the board had adopted detailed rules for the products, the industry would need time to arrange for packaging and establishing prices.
 
In recent years dairy companies have been marketing “half and half” in Chicago and other parts of the country.
 
9 January 1955, New York (NY) Times, “More Milk and Less Cream” by Jane Nickerson, pg. SM54:
“Half and half”—a mixture of milk and cream that might be called “a dieter’s compromise for coffee”—is 12 per cent fat (against 18 per cent for light cream and 40 per cent for whipping cream). It is high in calories in comparison with skim milk, but not nearly so high as whipping cream. One tablespoon of the latter contains fifty calories.
 
28 April 1955, New York (NY) Times, pg. 15:
LOW SODIM MILK
“Half-and-Half” Product Also
Is Approved by City
The manufacture and sale of low sodium and “half-and-half” milk products have been approved by the Board of Health, the Health Department announced yesterday.
(...)
The “half-and-half” product is a combination of milk and cream. Milk contains 3.3 per cent butterfat and light cream 18 per cent. The new product, with a butterfat content of 10 per cent, is expected to be used first in restaurants.
 
12 May 1955, New York (NY) Times, pg. 25:
Half-and-Half.
Milk Product,
In Shops Soon
     
21 June 1955, Dallas (TX) Morning News, part 2, pg. 3 ad:
Everybody Loves Half and Half
You can’t please everybody? Oh yes, you can, with Lucerne Half and Half. It’s half cream, half milk. You’ve been using milk on fresh berries? See what a big difference in enjoyment Lucerne Half and Half, at Safeway, makes. On cereal, too. Suits our family to a T. The children like the creamy flavor better than milk, and they need the extra richness. We adults, with an eye on the scales, get creamy taste without extra calories. It’s an excellent “cream” for coffee. I find Lucerne Half and Half a specially fine product. Only Grade A cream and milk are used. Homogenized so the cream is evenly distributed through the carton. You’ll like.