
Apparently the effect is named after a particular image, that appeared, with variations, on the tins and boxes of Droste cocoa powder, one of the main Dutch brands. It displays a nurse carrying a serving tray with a cup of hot chocolate and a box of the same brand. The recursive effect first appeared in 1904, and was maintained for decades, becoming a household notion. Reportedly, poet and columnist Nico Scheepmaker introduced wider usage of the term in the late 1970s.
I googled Droste and I discovered that it's a German surname that means Head Servant. And that's precisely what the packaging depicts.
Information and image courtesy: Wikipedia.