The origin of some golfing terms - "bogey" for example - are well documented, while terms like "albatross" and "eagle" are playful avian extensions of "birdie". The etymology of the word itself, though, is lost to history, even though it is a fairly recent term.
There is a hoary old tale of a probably mythical Colonel Bird, while the Times etymologist Phillip Howard believes the word dates from the beginning of the century - when much golfing terminology was fixed - from the older American slang of "bird" for somebody or something smart or accomplished.
Another possibility is suggested by a usage cited in the Oxford English Dictionary Supplement from Maclean's Magazine in 1911: "Lanesborough followed with a `bird' straight down the course, about two hundred and fifteen yards." Perhaps, then the word simply refers to the ball in flight, with a diminutive ending for particularly good examples, i.e. those that end in a stroke gained.