Interesting thread! This has me breaking out my Dictionary of New Zealand English, ed. by Harry Orsman.
Across in NZ the earliest printing he located of 'tinny' meaning 'lucky' was 1918 in an NZEF army newspaper: "Remarks are heard on the 'tinny' luck of the [poker-]players, while the 'stiffs' bemoan their luck."
So maybe NZ soldiers picked it up from Aussies during WW1 (?). Then again, 'The Bulletin' was always widely read in New Zealand in those days. They also printed verse by many poets from here.
According to Orsman, 'tin-arse' and 'tin-bum' came later to New Zealand, the earliest printing of either he found was 1984, but of course it may go further back in terms of popular usage.
As far as the derivation goes, he agrees with the whole tin=money theory. He also mentions that some folks have considered the word was a corruption of "[des]tiny", but discounts this immediately, so I don't really know the details on that.