There was a George George who died in 1916 at Edeowie, north of Hawker, in the Flinders Ranges of South Australia. He claimed that he was a cousin of Lloyd George - at the time Minister for Munitions.
George George was born in Wales c1840 (based on his age, believed to be 76 when he died in 1916). He lived at Edeowie from 1892 to 1916.
George George was believed to have a nephew, Edward George, working in Queensland as a drover (1916).
George was not believed to be married, although I have no real evidence as to whether he was or not - simply the belief of the locals who provided information to the policeman reporting his death (Edeowie was a pretty remote place, with very few people living within 30 miles or more).
I'm not related - just doing historical research in the area. I would be pleased to hear of any connections such as he claimed. If he or his nephew are related to you, I'd be happy to provide more snippets of information about George.