According to the National Museum of the Royal New Zealand Navy, in 1731 rum was made an official issue to seamen and the daily half pint was issued in two equal parts, one in the morning and the other in the evening.  This was neat spirit and drunkenness became rife especially on the West Indies Station.

This situation led to the following order (Edward Vernon was born in 1684 and died in 1757):

By Edward Vernon, Esq., Vice-Admiral of the Blue, and Commander-in-Chief of His Majesty’s Ships and Vessels in the West Indies.
Whereas it manifestly appears by the return made to my general order of the 4^th of August, to be the unanimous opinion of both Captains and Surgeons that the pernicious custom of the seamen drinking their allowance of rum in drams, and often at once, is attended with many fatal effects to their morals as well as their health, which are visibly impaired thereby, and many of their lives shortened by it, besides the ill consequences arising from stupifying [sic] their rational qualities which makes them heedlessly slaves to every brutish passion, and which having their unanimous opinion cannot be better remedied than by the ordering their half pint of rum to be daily mixed with a quart of water, which they that are good husbands may from the savings of their salt provisions and bread purchase sugar and limes to make more palatable to them. You are, therefore, hereby required and directed, as you tender both the Spiritual and Temporal Welfare of His Majesty’s subjects and preserving sobriety and good discipline in His Majesty’s service, to take particular care that Rum be no longer served in species to any of the Ship’s company under your command, but that their respective daily allowance of half a pint a man for all your Officers and Ship’s company be every day mixed with the proportion of a quart of water to every half-pint of rum to be mixed in one scuttled butt kept for that purpose, and to be done upon deck, and in the presence of the Lieutenant of the watch, who is to take particular care to see that the men are not defrauded in having their full allowance of Rum, and when so mixed it is to be served to them at two servings in the day, the one between the hours of 10 and 12 in the morning, and the other between four and six in the afternoon. And you are to take care to have other scuttled butts to air and sweeten their water for their drinking at other times, and to give strict charge to your lieutenants in their respective watches to be very careful to prevent any Rum or other spirituous liquors being privately conveyed on board the ship by your own boats, or any others, as both you and they must expect to answer for the ill consequences that may result from any negligence in the due execution of these orders. For which this shall be your warrant. Given under my hand on board H.M. ship the [Burford, 21st August, 1740.]

The order exists at the Public Record Office only in the form of a copy, which is neither signed nor dated. It is endorsed: “Copy of an Order for preventing Drunkness [sic], 21st Aug., 1740”, and with a note that it was enclosed “in Mr. Vernon’s of 7th Oct., 1740” (source: L. G. Carr Laughton – The Mariner’s Mirror, 1919).

The word grog therefore is said to be short for grogram, and to have been applied first as a personal nickname to Vernon, from the fact of his wearing a grogram cloak, and afterwards transferred to the mixture which he ordered to be served out. This has all the appearances of a folk etymology, but no evidence for grog prior to Vernon’s 1740 order has been found, and all the contemporary examples seem to support this origin. For the amusement of his shipmates, Thomas Trotter (1760-1832), Scottish physician to the fleet, wrote in 1781 The Origin of Grog, which celebrated the introduction of this mixture on the Burford some 40 years earlier:

(1829 edition)

Written on board the Berwick, a few days before Admiral Parker’s engagement with the Dutch fleet, on the 5^th of August, 1781. By Dr. Trotter.

’Tis sung on proud Olympus hill
The Muses bear record,
Ere half the gods had drank [sic] their fill
The sacred nectar sour’d.

At Neptune’s toast the bumper stood,
Britannia crown’d the cup;
A thousand Nereids from the flood
Attend to serve it up.

“This nauseous juice,” the monarch cries,
“Thou darling child of fame,
“Tho’ it each earthly clime denies,
“Shall never bathe thy name.

“Ye azure tribes that rule the sea,
“And rise at my command,
“Bid Vernon mix a draught for me
“To toast his native land.”

Swift o’er the waves the Nereids flew,
Where Vernon’s flag appear’d;
Around the shores they sung “True Blue,” ¹
And Britain’s hero cheer’d.

A mighty bowl on deck he drew,
And filled it to the brink;
Such drank the Burford’s gallant crew, ²
And such the gods shall drink.

The sacred robe which Vernon wore ³
Was drenched within the same;
And hence his virtues guard our shore,
And Grog derives its name.

¹ A favourite Song.
² Flag-ship at the taking of Porto Bello.
³ Admiral Vernon usually wore a Grogram cloak in bad weather, from which the sailors called him Old Grog; hence the name, in honor of him, was transferred to the spirit and water, because he was the first officer who ordered it in this manner on board his Majesty’s ships.

The following definition of grog is from A Classical Dictionary of the Vulgar Tongue (2^nd edition – London, 1788), by the English antiquary and lexicographer Francis Grose (1731-91):

Rum and water. Grog was first introduced into the navy, about the year 1740, by Admiral Vernon, to prevent the sailors intoxicating themselves with their allowance of rum or spirits. Groggy, or grogified; drunk.

The earliest instance of grog that I have found is from The Derby Mercury (Derby, Derbyshire, England) of Friday 3^rd February 1748:

SATURDAY’s POST.

From the WHITEHALL and GENERAL Evening Posts, &c. February 2.
The following is an exact Account of the late Action fought between Admiral Knowles and the Spanish Admiral; taken from the Jamaica Gazette.

Kingston in JAMAICA.

[…]
The next Day we met a Spanish Sloop form Cadiz, going into the Havannah, who told us of the Peace: I cursed him for coming in our Way, for we should have gone and taken all the Galleons else, and been as rich as Princes: I am sure we deserved it, for we lived at Short Allowance all the Cruize, and but two Quarts of Water a Day, to make it hold out in Hopes of meeting them (but short Allowance of Grog was worst of all) and now we have bro’t this Prize here, we are told she will be given up to the Spaniards again, so we have fought them for nothing.