1752, supposedly a facetious formation from gaze + -bo, Latin first person singular future tense suffix (cf. videbo "I shall see"), on model of earlier belvedere "cupola," from Italian bello verde "handsome sight." But perhaps rather a corruption of some oriental word.
One gazebo wants to buy a castle in the old country; another wants a racing stable; another a steam yacht.
And if I find any gazebo getting too thick with her, then up speaks little Bertie for the word that makes her his.
The Folly was a long two-storied house, with a tower or gazebo at one end.