In 1858 when he was surveying and exploring the region north of Lake Torrens, Benjamin Babbage mapped a lake that he named Lake Gregory, after Augustus Charles Gregory, his second-in-command. This was just part of the process of disproving Edward Eyre's horseshoe-shaped barrier of salt lakes which was known as Lake Torrens and which appeared to bar progress further north. The combined explorations of Babbage, Warburton, Stuart and Gregory in 1858 showed that there were substantial gaps through the arc of lakes from Lake Gairdner and Lake Torrens in the west, Lake Eyre in the north and Lakes Gregory, Callabonna and Frome in the east.
Babbage's Lake Gregory is now Lake Eyre South, Governor Richard MacDonnell naming it Lake Eyre in 1860 subsequent to his tour of the northern regions. In a lecture he gave in 1863 and published as Australia: what it is and what it may be MacDonnell said 'the main central portion of the old lake which I have called Lake Eyre after its first discoverer'. Eyre Lookout on its southern shore is the point that Edward Eyre reached on 14 August 1840.
There is a Lake Gregory further to the east (28 degrees 91' 139 degrees 05'E) that Augustus Charles Gregory discovered and named in June 1858. He had been exploring west from Queensland looking for the missing explorer Leichhardt, and coming down the Strzelecki Creek found the gaps in the horseshoe lake. Briefly then there were two Lake Gregorys adjacent to each other in 1858-59. Lake Eyre (north and south) is Australia's largest lake, fed by an enormous catchment that drains an area of 1.3 million square kilometres. Lake Eyre however is usually dry, flooding infrequently and filling even more rarely.
Further reading:
Manning, Geoffrey H. Manning's place names of South Australia, [Adelaide] : G.H. Manning, 1990
Bonython, Warren. The Great filling of Lake Eyre in 1974, Adelaide: Royal Geographical Society of Australasia, South Australian Branch, 1989