My giddy aunt
An exclamation of surprise.
The word giddy has been used to mean mad or stupid since the first millennium. The Old English word gidi derives from the Old Teutonic word for God - gudo. So, those who were labelled giddy were those who were possessed of God. The more recent (16th century) use of dizzy, to mean or affected with vertigo began life as the word turngiddy. Shakespeare alluded to this in Lucrece, 1593:
To spoil antiquities of hammer'd steel,
And turn the giddy round of Fortune's wheel.
The link between the two meanings of giddy are exemplified by the Sufi devotees - the Whirling Dervishes (properly called the Mevlevi).
There's nothing especially giddy about aunts. The word has been applied as an intensifier to all sorts of things - giddy-headed, the giddy ox, etc. Shakespeare used the word 30 or more times in his plays and associated it with both Goths and geese. He seems to have noticed that giddy works best with other 'G' words, although he missed out on the only other 'giddy' phrase to have lasted apart from giddy aunt, that is, the giddy goat.
While it is the 'giddy goat' it is always my 'Giddy aunt!', that is, it is used as an exclamation rather than a description. The first use I can find of 'my giddy aunt' is in The journal of a disappointed man, 1919, by W. N. P. Barbellion (the pseudonym of the English author Bruce Cummings).
Of course the giddiest aunt of them all is the character in Charley's Aunt - Brandon Thomas' farce, which was first performed in 1892. In the play Charley Whyckham's aunt is said to be "from where the nuts come from" (Brazil). The farce revolves around her impersonation by a man, but I can't find any reference in the play to 'her' as giddy.