To “tie one on” means to get drunk.
In The Wordsworth Book of Euphemisms, Eric Partridge suggests that this expression is derived from “hang one on” (circa 1935), which originated in the US. It is certainly clear that a “hangover” is the miserable memento of having hung or tied one on.
Some sites claim that “tie one on” dates back to the wild west in the US, in the 1800s, where a cowboy would have to tie up his horse to a hitching post before he could go into the saloon and get drunk. I think that’s stretching much too far for a connection.
Wikipedia says that to “tie one on” means to drink for the purpose of getting drunk, especially when one is currently still drunk or hungover from a previous drinking session, thus having something to “tie one on” to. I think that’s also stretching too far.
The Oxford English Dictionary compares “tie one on” to the British slang phrase “tie a bun on,” also meaning “to get drunk.” This phrase appeared around 1901. There was a theory that being able to walk with a bun balanced on your head was a way of proving you were sober, much like walking a straight line today. The theory continued with this explanation: if you were drunk, you would tie the bun to your head so it wouldn’t fall off. Maybe, if you fell down, the bun would serve as a pillow.
There is, however, another US phrase, appearing around the 1940s, that means to get drunk: “tie a bag on.” Partridge’s A Dictionary of Slang lists “Bag: a pot of beer, from 1887.” The search continues to the phrase, “get/put (one’s) head in a bag,” probably from horses’ nose bags or feed bags.
It’s not a big stretch to connect a horse’s feed bag to someone drinking out of a pot of beer as if it were a feed bag tied around his neck. This seems confirmed by the expression “in the bag,” used to mean drunk, also “half in the bag” (1920s) and “have a bag on” (1940s and still in use).
We always tend to abbreviate phrases and words. “Tie one on” is easier to say than “tie a bag on,” especially if you’re already half in the bag!