Tight as a boiled owl means completely drunk. Tight already meant drunk; the boiled owl is there to make it absolute.

It is on lists of Victorian slang, and J. Redding Ware did put it in Passing English of the Victorian Era (1909): Tight as a biled (or boiled) owl (American). Completely drunk. Note his label — American. The phrase was a visitor.

The owl had been drunk for a long time before it was boiled. Green's Dictionary of Slang finds as drunk as an owl in the Gentleman's Magazine in 1770. The boiling seems to have happened in America: the Dictionary of American Regional English has drunk as a biled owl in an Indiana newspaper in 1848, and the humorist Artemus Ward used biled owl in 1862.

It crossed the Atlantic quickly. Green's has the London journalist George Augustus Sala writing as lushy as a boiled owl in his 1864 novel Quite Alone, then tight as a boiled owl in the Fifeshire Journal in 1882, and in Ally Sloper's Half Holiday, the London comic paper, in 1891.

Why an owl, and why boiled? Ware gave up: Here there is no common sense whatever, nor fun, wit, nor anything but absurdity. He quoted a newspaper writer of December 1892 who protested that the phrase was a gross libel upon a highly respectable teetotal bird which, even in its unboiled state, drinks nothing stronger than rain-water.

Ware did float one guess of his own: that, like many puzzling proverbs, it might be a proper name mangled out of recognition — perhaps drunk as Abel Doyle. He offered no evidence for it, and nobody since has found any for that or any other explanation. The honest answer is still: nobody knows. The absurdity may be the whole point: drunk enough that only a ridiculous comparison will do.