In northern Canada – Nunavut and the Northwest Territories – there are two islands, Melville and Bathurst, side by side. In northern Australia – and I mean Northern Australia – one can also find Melville Island and Bathurst Island, right next to each other. I’m wondering, do they take their names from the same source, or is this just coincidence?
I don’t know the answer to your question, but I felt I had to post because I happen to have a map of Melville and Bathurst Islands (the Canadian ones) on my desk today, and was looking at it five minutes ago. What a strange coincidence.
They do not look like a vacation paradise, unless you like being cold and alone or you want to look for the north magnetic pole and watch your compass get confused. For what it’s worth, a caption on the map says “Permafrost has been measured to a depth of 1,800 feet on Melville Island.”
So there.
Perhaps Messrs. Melville & Bathurst asked for a refund after they found out where “their” islands were. 
Hey, fiddlesticks, we can’t all be Prince Edward, Saint Thomas, or Joey Madagascar.
From the Britannica:
The Royal Navy was pretty well exploring the ends of the Earth in 1818-19, and naming things after the Secretary for War and the Colonies and for the First Lord of the Admiralty was evidently the In-Thing.
The Australian Melville Island was also named (like the Australian Bathurst Island) by Captain Phillip Parker King–my typo there…It might appear that Sir William was less original than Captain King by a year, but with travel and communications times in those days, I imagine they both independently hit on those particular bits of flattery.
My source is http://www.britannica.com/.
The Canadian islands were discovered in 1819 by Sir William Parry. They were uninhabited.
The Australian islands were 1st spotted in 1644 by Dutch navigator Abel Tasman and were named in 1818 by 1818 by Captain Phillip Parker King. They were occupied by the indigenous Tiwi peoples.
The islands have the same eponym: Robert Saunders Dundas, 2nd Viscount Melville, then first lord of the Admiralty,and Henry Bathurst, the 3rd Earl Bathurst, then secretary for war and the colonies.
Those 19th century Brits got around.
MEBuckner not only beats me to the answer, he puts the link in scripted format.
MUAHAHAHA! Once again I prove my superiority!
[sub]Oh, did I say that out loud?[/sub]