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The informal phrase (here’s) mud in your eye is a humorous drinking toast.

—Cf. also the phrases here’s looking at you and to wet the other eye.

The origin of the phrase (here’s) mud in your eye is unknown. In my opinion, the image is perhaps simply that mud in one’s eye blurs one’s vision like alcohol does.

The following song (unrelated to the drinking toast) uses the image of mud in one’s eye to characterise a baseball umpire accused of being visually impaired—from an advertisement for this song, published in The Detroit Free Press (Detroit, Michigan, USA) of Sunday 9^th May 1909 [page 23]:

Here’s the Chorus of “Let’s Get The Umpire’s Goat”
Let’s get the Umpire’s goat, goat, goat, let’s make him go up in the air.
We’ll yell Oh, you robber! Go somewhere and die—
Back to the bush—you’ve got mud in your eye.
Oh, what an awful decision! Why don’t you put spectacles on?
Let’s holler like sin and our side will win, when the umpire’s nanny is gone.

—Cf. also the phrase where’s your white stick?, used to express disagreement with the referee during a match.

These are, in chronological order, the earliest occurrences of the phrase (here’s) mud in your eye that I have found:

1-: From the Oregon Daily Journal (Portland, Oregon, USA) of Friday 18^th August 1911 [page 9, column 1]:

Salutes Bystanders; Drinks Acid.—“Here’s mud in your eye!” shouted G. E. Mitchell, aged 31 years, 7123 Forty-first avenue southeast, at 5:30 o’clock last night, as he swallowed the contents of a vial of carbolic acid in a drug store at Grand avenue and Hawthorne, before a half dozen witnesses and the clerk could knock the deadly fluid from his hands. Twenty minutes later, while on the way to the hospital In a Red Cross ambulance, the man died, although Patrolmen Fuller and Adams, who had been summoned to the drug store, had poured a quantity of olive oil down his throat. The reason for the act has not been determined, although his father, R. S. Mitchell, 772 Hoyt street, declares he had been despondent over inability to secure work. A wife and two children, aged 5 and 8 years, survive him.

2-: From Grip Knights Weekly Dope, published in The Shreveport Times (Shreveport, Louisiana, USA) of Monday 15^th January 1912 [page 2, column 2]:

“Here is mud in your eye,” is the logical toast now.

3-: From You Never Can Tell About a Woman, by Maverick Terrell (1875-1934) and H. O. Stechhan, published in The Smart Set: A Magazine of Cleverness (New York City, New York, USA) of August 1912 [Vol. 37, No. 4, page 138, column 1]:

Nell
You from Texas and can’t stand one shot of corn juice—just among friends? Come on. […]
Wedgewood
Well, just two fingers—to be sociable, sister—to show there’s no hard feelings.
Nell
Good! (She slaps him vigorously on the back, turning him completely around, and holding her own glass high, as if pledging good fellowship.) That’s right, old man—here’s mud in your eye an’ a quick trip back to Texas!

4-:From The Chambers Bugle (Chambers, Nebraska, USA) of Thursday 2^nd December 1915 [page 1, column 2]—although, here, the phrase is perhaps not the drinking toast:

Everybody is familiar with the saying, “Here’s Mud in Your Eyes.” So it is these windy days in Holt, “Here’s Sand in Your Eyes.”

5-: From Frivolous Feed For Fans, by Tom Martin, published in The Wisconsin State Journal (Madison, Wisconsin, USA) of Friday 19^th May 1916 [page 13, column 4]:

The State Boxing Commission suspended Freddie Welsh because he had liquor in his corner during his fight with Ever Hammer at Kenosha. Welsh claims the stuff in the bottle was tea.
AND—
“Lips that touch licker
Shall never touch mine,”
Sings the boxing commish
With a harmony fine.
“That may be all right,”
Chants Welsh in reply,
“But I must have my tea—
Here’s mud in your eye.”

6-: From The Treating System (If Women Indulged), published in Puck: America’s Cleverest Weekly (New York City, New York, USA) of Saturday 10^th February 1917 [page 10, column 2]—Scene: The Corner drugstore:

Leonora: Well, girls; here’s hoping you never see the back of your necks.
Myrtle: Happy days, old dears.
Genevieve: Shoot!
(They drink.)
Myrtle: Well, I guess I’ll buy a little drink. Name your poison.
Genevieve (giggling): Make mine a vanilla in-and-out, with grated cinnamon.
Leonora: I might take a shot at that.
Myrtle: It sounds pretty vicious. Let ’em roll, George.
(George lets ’em roll)
Myrtle: Well, here’s hoping.
Leonora: Here’s to your beautiful eyes, Myrt.
Genevieve: Yes, here’s mud in your eye, Myrt, old scout!
(They drink.)