Update, 2021: DNA testing has now linked me to other families descended from David STIBBLES and Elizabeth BARNET of Dundee, so they are definitely the parents of John. He is no longer a mystery!

John Edward BARNETT is still something of a mystery to me. At his marriage in Morpeth NSW on 22 January 1857 to Elizabeth LOGUE, he stated his birthplace as Brooklyn, America, and his occupation as mariner. His parents' names were given as David BARNET, horse dealer, and Elizabeth STIBBLES or STAPLES. On the birth registration of his son Charles, his birthplace is stated as Brookton, USA (the clerk also wrote his wife's maiden name as LOWE, so perhaps there was a problem with her Irish brogue). There is no record of a John BARNETT arriving in NSW, until we recall that the 1850s were the time of the Gold Rush in Australia, when people were doing almost anything to get here. Ships' Deserters 1852-1900 compiled by Jim Melton, p.31, led me to the NSW Government Gazette, 4 Mar 1853, p.434

Return of seamen who have deserted from their respective
vessels, as reported at the Water Police Office:-

Edward Barnett, [ship] Milbourne, seaman
W.C.Mayne, Inspector-General of Police
Sydney, 4th March, 1853.

This could explain a change of name from Edward to John (Edward), and would explain why he worked on the docks until the day he died. John's death occurred (as I found from his gravestone at Rookwood cemetery), when he

was accidentally
killed on the S.S. Ranelagh
1st May 1882
aged 47 years

As well as death and funeral notices, the Sydney Morning Herald of 2 May 1882 carried a brief news item about his death:

A fatal accident occurred at 10 o'clock yesterday morning to a wharf labourer, named John Barnett, whilst at work unloading near the fore-hatch, on board the Ranelagh, lying at the A.S.N. Company's wharf, Circular Quay. He was making the staging fast when the part on which he was leaning canted over, and he fell down the hold, a distance of 20 feet, striking his back on a crossbeam. He was immediately brought to the deck and examined by Dr. Ewen, who pronounced life to be extinct. The body was conveyed to the deadhouse, Circular Quay, where it awaits an inquest. The unfortunate man is about 56 years of age, married, and leaves a wife and nine children.

[Even then, newspapers got the facts wrong in their haste for a story].

The inquest mentioned above was held, and created a new puzzle:

Name of deceased - Barnett, John
Age - about 50 years
Where born - Dundee, Scotland
Locality where death occurred - On board the steamship Ranelagh, lying at the A.S.N. Company's Wharf, Sydney.
Verdict or cause of death - Effects of injuries accidentally received on board the steamship Ranelagh.

The difference in his age is reasonable, and errors could be explained by the details being given by his workmates, rather than family members, but 'Dundee, Scotland' is rather particular to be a mistake. Also, my research has turned up only eight children to John and Elizabeth, not nine:

1. Charles Edward John, b. 24 January 1858 Sydney
2. Emily Jane, b. 1861 Morpeth
3. Frances,b. 1864 Morpeth
4. Edith, b. 1866 Morpeth
5. Clara, b. 1869 Morpeth
6. Ernest, b. 1872 Newcastle
7. Arthur Henry, b. 1875 Sydney
8. Frederick William, b. 1878 Sydney

His widow Elizabeth died some years later at Newington Asylum, on 1 January 1910, and was buried at Rookwood with her husband.

I have now found a family in the Scottish Old Parish Registers, as follows, which appears to be relevant:

David STIBBLES married Elizabeth BARNET 24 Feb 1825, Dundee, Angus, Scotland. Their children were:

1. John bap. 21 Aug 1825 Dundee
2. Elizabeth bap.31 Jan 1830 Dundee
3. Agnes bap. 30 Dec 1832 Dundee
4. Elizabeth bap. 14 Feb 1836 Dundee
5. David Henry bap. 17 Dec 1837 Dundee
6. James bap. 12 Dec 1841 Dundee

So, my great-great-grandfather John BARNETT appears to be Edward BARNETT, shipjumper, a son of David STIBBLES and Elizabeth BARNET.