The places in Victoria are named after Lieutenant-Governor Charles Joseph La Trobe (20 March 1801 - 4 December 1875) who arrived in Melbourne in 1839 where he had been appointed Superintendent of the Port Philip District.
His biog says he "was born in London of Moravian parents and had made his reputation as a mountaineer, adventurer and writer". He died in England in 1875.

Elizabeth's parents might just have liked the look of the name, and/or perhaps her father as a master mariner had seen the name on buildings and other places in Victoria - or perhaps they were related to the La Trobe family.

Liz, as Elizabeth was well known in Douglas wouldn't there have been a newspaper obit./ M.I.?

Sue