Having researched the Brownhall Hamilton’s for nigh on twenty years I can safely say there is no link between those of Donegal and Ballarat, Victoria. There were several saints in Ireland named St Ernan and it was common for the Irish, expats included, to name their homes after a saint. This is what John Hamilton of Brownhall did with his house St Ernan’s and no doubt your ancestor did the same thing.

John Hamilton’s grandson – my GG Grandfather Augustus Courbarron – named his own house in Sydney ‘St Ernans’ ( a family thing )even though it did not resemble the original in Ireland.

I can make a few more enquiries however just in case there is something I have not covered – will post it here.