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Originally and chiefly British, the dated slang phrase to wet the other eye (also to wet t’other eye) means: to have one alcoholic drink after another.

The image is that the first drink wets one eye, and the second drink wets the other eye—cf. an extended form of the phrase in quotation 4 below.

This phrase occurs, for example, in the column How’s About, by ‘John’, published in the Mid-Week Free Press (High Wycombe, Buckinghamshire, England) of Wednesday 14^th March 1973 [page 8, column 9]:

IT’S A HARD life. There we were plastered in good old Buckinghamshire mud the other Sunday morn, descending on a hostelry tucked away in the woods for a welcome pint, well-earned and well walked for.
But when we arrived at this haven, we were not alone, far from it. There was a fine assortment of rumps propped up before the bar, rumps which had to be rounded before one could get close to the counter. These said rumps were the property of the ‘old noggin’ types.
[…]
[…] I managed to round the rumps and get the message through, three pints and a lemonade.
[…]
The bottoms round the bar were after their second ‘old noggin’ by now […].
[…]
We too were ready to wet the other eye.

—Cf. also the informal phrase (here’s) mud in your eye, which is a humorous drinking toast.

These are, in chronological order, the earliest occurrences of the phrase to wet the other eye (also to wet t’other eye) that I have found:

1-: From The Life and Adventures of Bampfylde-Moore Carew, the noted Devonshire Stroller and Dog-Stealer; as related by Himself, during his Passage to the Plantations in America. Containing, a great Variety of remarkable Transactions in a vagrant Course of Life, which he followed for the Space of Thirty Years and upwards (Exon: Printed by the Farleys, for Joseph Drew, 1745) [page 89]:

Having dres’d himself in a Sailor’s Habit, he went to Exeter-Key, and slily enquired for some of the King’s Officers, whom he informed that he belonged to a Vessel that came from France, and which had a very large Quantity of Run-Goods of great Value on board; that the Captain and he had differ’d, and —— Here he was about to proceed, but the Officers (with great Joy and Attention heard what he had said, concluding the Sequel to be very favourable to them) interupted [sic] him, taking him into the Custom-House, and filled him out a Bumper of Cherry Brandy, which when he had drank they forced another upon him, persuading him to wet the other Eye.

2-: From The Old Women Weatherwise, published in Analects in Verse and Prose, chiefly Dramatical, Satirical, and Pastoral (London: Printed for P. Shatwell; J. Dodsley; and T. Davies. 1770), by George Saville Carey (1743-1807) [Vol. 2, page 169]:

SONG.

TWITCH, with the bottle in her hand.
Wet the other eye,
Wet the other eye,
Let’s be jolly,
Melancholly
Is a folly;
Then refrain—
’Tis in vain
To complain,
Let us wet the other eye. [drinks.

3-: From The Committee (London: Printed for John Bell, 1776), a comedy by the English playwright Robert Howard (1626-1698) [Act 4, page 57]:
—Note: The phrase occurs neither in the original text—i.e., The Committee (London: Printed for Henry Herringman, 1665)—nor in editions anterior to 1776 such as The Committee: Or, the Faithful Irishman (London: Printed for Jacob Tonson, 1735):

Teag. I am a cup too low.
Car. Here then. [Gives him a glass.]
Teag. I should like to wet t’other eye.
Car. Here?

4-: From an account of a court case, published in The Reading Mercury, Oxford Gazette, and General Advertiser of Berks, Bucks, Hants, Oxon, Surrey, Sussex, and Wilts (Reading, Berkshire, England) of Monday 30^th June 1794 [page 3, column 2]:
—The plaintiff was Mr. Woodgate, and the defendants, Mr. Hitchens and Mr. Hilder, were both attorneys:

LONDON, Saturday, June 28.
[…]

THE KING v. HITCHENS and HILDER.

[…] A person of the name of Wells summoned Mr. Woodgate before a justice of peace, for having a cart without a name upon it. The magistrate saw through the business, and discharged the complaint. Wells not being satisfied with this, next applied to Lawyer Hitchens, who said he would put him in a way of obtaining substantial justice. For this purpose, Hitchens procured a man of the name of Johnson, a day-labourer, to commence an action against Mr. Woodgate, to recover the penalty. Hitchins [sic] and Hilder gave Johnson one dram to wet one eye, a second dram to wet the other eye, and, last of all, they gave him a dram to wet his nose. This distribution of justice by Lawyer Hitchens, made this poor fellow completely drunk; and, when he was in a state of intoxication, they procured his mark (for he could not write) that he would bring the action.