…as in “To get drunk”?
I assume that you’ve done all the Googling and have found, like me, that there appears to be no consensus on the origin.
My favorite (almost certainly) apocryphal version found in the comments of one site refers to the British “tie a bun on” mentioned as a possible earlier version. The commenter suggests that one proved their ability to drive their horse and carriage home by walking through the pub with a bun balanced on their head. Those who were unable to do so had “to tie the bun on”.
I think it refers to a wing, as in “you can’t fly on one wing”, an old expression used when urging someone to have another drink.
I like the Fred Flintstone line: “I’m going to tie one on–a feedbag, that is!”