Since Nick Thieberger is now resident in our department at UHM, the Australian name struck me as worthy of a little research. Nick is from Melbourne, and has worked with a number of Australian languages as well as his current research on South Efate in Vanuatu. He recommended an interesting little book, Australian Aboriginal words in English: their origin and meaning by R.M.W. Dixon, W.S. Ramson, and Mandy Thomas. It’s a neat little book, meant to professionally counter the proliferation of inaccurate etymologies of the Australian lexicon. Here’s the entry on “budgerigar” (p. 89).
Also betcherrygah, betshiregah, and budgerygah.
[Possibly mispronunciation of Kamilaroi, eastern New South Wales gijirrigaa.]
The small green and yellow parrot Melopsittacus undulatus, occuring in drier mainland areas, often in large flocks. The budgerigar has become extremely popular throughout the world as a cage-bird. It is also called the love-bird, shell parrot, warbling grass parakeet and zebra parrot. The shortened form budgie is very common. 
The “mispronunciation” struck me as highly curious. Why would English speakers, who have a perfectly good voiced velar stop in their phoneme inventories, make such a peculiar substitution of [b] for [g]? I figured it would be worth doing a little more looking. I then stumbled upon this thought provoking entry in the back of the very same book (p. 205).
budgeree /ˈbʌdʒəri/ /ˈbʊdʒəri/
Also with much variety, as boodgery, boojeri, boojery, and budgeri. Australian pidgin.
[Dharuk, Sydney region, adjective bujiri ‘good, right’.]
Good, pretty, fine. 
It doesn’t take much consideration to see that a blend of the two terms gijirigaa and bujiri is easily possible here, what with sharing two syllables. My suspicion is that the folk etymology “good eats” arose after the blend formed, and people familiar with the term “budgeree” applied its meaning. The reason for the blend arising in the first place remains obscure.