The story of the lamington on the tea towel is a nice one, but sadly it's one of a great many myths about their creation.
Some years ago I researched the history of the lamington, checking written records back to 1830, and appealing on radio shows around Australia for any early lamington recipes, or letters or menus or newspaper accounts that mentioned them.
After an enormous response it appeared that lamingtons were invented in Brisbane around the early 1900s, probably by Amy Shauer who taught cooking at Brisbane central Technical College from 1895 to 1937. She also wrote three very popular cook books, and developed cookery courses for schools and colleges across Queensland, and was a famous cake maker and cake judge at Shows.
It's likely the first lamingtons were invented in Amy Shauer's cooking class and named after Lady Lamington, who was the school's patroness and extremely interested in education for girls. (One elderly correspondent, who remembered those days well, informed me that Lord Lamington was a pompous ass, and that no one would ever have named a cake after him. But Lady Lamington was much loved.)
The earliest lamingtons - those in all recipes recorded before 1910 - were for one big single chocolate and coconut covered cake, not the little lamingtons 'cut into oblongs' we know today. So any story that describes the invention of small cakes- as we know the lamington today- is probably a myth. The original lamingtons were also made with a butter cake base, not sponge cake.
By the 1920s lamingtons - both small and large, but not yet with any cream in them or- horrors- pink icing instead of chocolate- were one of Australia's favourite cakes, for a very good reason. In a hot dry climate, before the days of air conditioning and insulation, most cake turned into rock a few hours after you'd taken it out of the oven. Rolling cake in moist chocolate sauce then rolling it all in coconut (traditionally using two knitting needles to hold it) was a great way to keep a cake tasting soft and fresh for days.
PS It still is.