Green’s Dictionary of Slang

drop in one’s eye n.

[? SE drop of liquor, or one is on the verge of drunken tears]

[late 17C–18C] a state of (near) drunkenness.

In phrases

have a drop in one’s eye (v.)

[late 17C–mid-19C] to be tipsy.

take a drop in the eye (v.)

[late 17C–early 18C] to have a drink.