There seems to be not much real scholarship on this subject, but the answer seems to be that the "rotating serving tray" was first invented in Britain in the late 18th century to reduce the need for hovering household servants at mealtimes, although it's not clear if this was due to high labor prices or changes in social mores towards greater privacy during meals as "family time". Made of wood or silver, they gain particular favor among the American upper classes during the early 20th century -- unlike contemporary Europeans, America gentry generally seem to have preferred servants to be waiting in the next room, rather than constantly hovering nearby.
The ubiquity of "Lazy Susans" in Chinese culture seems mostly to have been a 20th-century development, according to this Smithsonian article.. However, as the article explains, they became ubiquitous in modern American-Chinese resturants in the early 1960s as part of a set of deliberate marketin campagins to make Chinese resturants in the United States seem more "sanitary" and "modern".
I've found two popular-press articles that discuss the history and possible origin of the name a little more:
http://latimesblogs.latimes.com/home_blog/2010/03/lazy-susan-history-who-invented-mystery.html