The dated jocular exclamations my giddy aunt!, my sainted aunt!, etc., express surprise, consternation, etc.
They are extended forms of the exclamation my aunt!, which was used for example by the British novelist, short-story writer and poet Rudyard Kipling (1865-1936) in The Story of the Gadsbys: A Tale without a Plot (Allahabad, India: A. H. Wheeler & Co., 1888):
Mrs. G. (piteously).—Oh, don’t make fun of me! Pip, you know what I mean. When you are reading one of those things about cavalry, by that idiotic Prince—why doesn’t he be a Prince, instead of a stable-boy?
Captain G.—Prince Kraft a stable-boy! Oh, my Aunt! Never mind, dear! You were going to say?
Mrs. G.—It doesn’t matter. You don’t care for what I say.
These are the earliest occurrences of the extended forms that I have found, in chronological order:
1-: my sainted aunt!—From Puzzledom, published in The Bristol Mercury (Bristol, England) of Saturday 11th January 1890:
THE EFFACEMENT OF SORCERER.
Sorcerer let go this deadly jargon:—
My 1 is a breath that will never do here;
My 2 is a liquid that’s found in a mere;
My 3 4 together are flowers, just try ’um,
And 4 and 5, watching, persistently eye ’um,
My whole is a gathering, such as, I vow,
We all of us seem to be making just now.
“My sainted aunt!” cried Percy Verance, “What
Can be an answer to such awful rot?”
2-: my sacred aunt!—From Epaulettes (Bousfield and Co., 1902), by Torin Blair—as quoted in The Daily News (London, England) of Saturday 22nd March 1902:
Where was Bing? It was the first time he had been under fire, and when he saw men dropping round him with yells and groans—and no enemy to be seen—nothing tangible to get at—no gallery to play to and horribly alone, he—well, no one quite knows what happened. The subaltern with him was badly hit, and when they found Bing he was crawling on the ground with staring eyes and a white, terror-stricken face, raving about “white-coated Cossacks.”
“My sacred aunt,” murmured the staff officer, in an undertone, “I’ve seen a good many cases of ‘nerves,’ bad ’uns at that, too, but this takes the blessed bun.”
3-: my giddy aunt!—From The Weekly Telegraph (Sheffield, Yorkshire, England) of Saturday 12th December 1903:
Tales of London Town
The Romance of a Pair of Socks.
[By E. Burrowes.]
Advertisement from the “Morning Post,” December 4th:—
Wanted.—Gentlemen’s underclothing to mend. Moderate charges; neat work; prompt execution. Socks a speciality. Apply R. D., Model Buildings, Mary-le-bone.”
Montague Rollestone threw down the paper in which he had just read the above advertisement, and thrust his hands through his hair with a rueful laugh.
“Jove!” he muttered, “the very thing. ‘Socks a speciality.’ My giddy aunt!”
4-: my holy aunt!—From the account of the case of Playfair v. Playfair and Affleck-Graves in the Divorce Division, published in The Lancashire Daily Post (Preston, Lancashire, England) of Thursday 19th March 1908:
Captain Dermot Affleck-Graves, the co-respondent, was next called. […]
Questioned about his letter to Mrs. Playfair, witness said a certain phrase was a phrase that had been used by the respondent’s little boy, and Mrs. Playfair wrote to him and told him about it. Mr. Grazebrook read a passage in which were the expressions:—
Oh, you little bit of Turkish delight! . . . You little duck! You little ripper! Barby, dear, now do you expect people to write calmly and prudently? . . . Oh my holy aunt!