The word in the title, pronounced ˈbɒbzidaɪ, is one I have only ever heard from New Zealanders. There is even a folk group of this name, but I am not sure if they are from NZ or Aus. You can see them in the picture. The word means “a great fuss” and is usually used in the phrases “to kick up bobsy-die” and “to play bobsy-die”.
According to the OED the word started out life as Bob’s-a-dying in the UK (first OED citation: 1829). Thomas Hardy seems to have been fond of the expression. He has two citations in the OED entry. It also appears in a work called Northumberland Words (1892), so it seems to have been used fairly widely in England at one time.
I wonder if the UK version is still in use anywhere and also where on earth it came from in the first place. The OED is silent on its origins.
Update: I have just consulted my copy of Brewer’s Dictionary of Phrase and Fable (1989, 14th ed.) Unfortunately, it does not list Bob’s-a-dying, but it does have an entry for bobbery, which also means “a great fuss”. The OED also lists this word and its etymology is:
According to Col. Yule, and others, an Anglo-Indian representation of Hindi Bāp re ! O father!, a common exclamation of surprise or grief. Forby has it in 1830 as East Anglian dialect; and it has been plausibly (as to the form) referred to Spanish boberia folly; but the evidence for its origination in India is decisive.
Curiouser and curiouser.