
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. Could be considered the drinking equivalent of chain smoking.

Some think that if you’d been drinking, to tie on one would be to have another drinking session before the hangover effects take hold, basically “tieing” one drinking session to the next “one”.

British slang for this term, “tie a bun on”, also means “to get drunk”. Around the 1870s, ‘a bun” was slang for getting drunk. Seems that an old-school sobriety test was to try to walk while balancing a bun on your head. If you were drunk the bun would fall off – unless you ‘tie the bun on’ to pass the test. So basically, to ‘tie one on’ is to get so drunk that you couldn’t walk with a free-standing bun on your head.

National Tie One On Day is a day for celebrating baked love every year on the Wednesday before Thanksgiving, on November 24 this year. On Thanksgiving Eve, apron-wearers all over America share some love with someone around them with a loaf of bread and an apron.
Further Reading
Sources
Urban Dictionary
The Meaningfulness of Little Things
Hot Idioms