In a Herald column last week, Colin James explored the possibility of electing the Governor-General. Like The Holden Republic, I am of two minds about this. My natural preference is for election, but generally you have elections when you want the holder of a position to be accountable to the public for their performance. Which implies that there is something they can be held accountable for. Given that the Governor-General is an almost entirely ceremonial position whose actions are almost completely dictated to it by our elected government, election seems constitutionally pointless, and may even result in a Governor-General trying to do things (such as, say, vetoing legislation - or worse, governments - which isn't really their job) on the grounds that they have an electoral mandate.
And OTOH, as Lewis points out, it works perfectly well for the Irish. They have an elected President who is essentially a "twink" Governor-General. So it essentially all comes down to political culture, and whether we would be able to establish one of Governors-General (or Presidents) who were neutral and impartial, as in Ireland.
Keith Locke's Head of State Referenda Bill [PDF] includes an option for direct election of the Governor-General. I guess we just have to hope that it gets drawn from the ballot.