It is difficult to say for sure who thought of the acronym. A number of accounts have been written.

One of the first occurrences of the word appeared in a book of sketches by Signaller Ellis Silas and in its foreword by Sir Ian Hamilton. Silas Ellis served with the AIF at ANZAC Cove.

Silas dedicated his book Crusading at ANZAC anno domini 1915, published in 1916 "to the honour and glory of my comrades with whom I spent those first terrible weeks at ANZAC".

In the foreword General Sir Ian Hamilton attributes to himself the coining of the acronym as a convenient piece of telegraphese devised for security purposes. He writes,

As the man who first seeking to save himself the trouble, omitted the five full stops and brazenly coined the word "ANZAC", I am glad to write a line or two in preface to sketches which may help to give currency to that token throughout the realms of glory.

The Official history of New Zealand's effort in the Great War, Volume 1: The New Zealanders at Gallipoli, quotes General Sir Ian Hamilton's account above and also gives an account by General Sir W. R. Birdwood.

...In the "ANZAC Book",General Birdwood stated that when he took over the command of the Australian and New Zealand Army Corps in Egypt, he was asked to select a telegraphic code address for his Army corps, and adopted the word"ANZAC". Later on after the landing, he was asked by General Headquarters to suggest a name for the beach, and in reply he christened it "ANZAC Cove".

The Australian war historian C.E.W. Bean attributed the acronym to a Lt A.T. White RASC of the British Regular Army.

One day early in 1915 Major C.M. Wagstaff, then a junior member of the "operations" section of Birdwood's staff, walked into the General Staff office and mentioned to the clerks that a convenient word was wanted as a code name for the Corps. The clerks had noticed the big initials on the cases outside their room - A. & N. Z. A. C. and a rubber stamp for registering correspondence had also been cut with the same initials. When Wagstaff mentioned the need of a code word, one of the clerks, (according to most accounts Lieutenant A.T. White) suggested: "How about ANZAC?" Major Wagstaff proposed the word to the general who approved of it, and "ANZAC" thereupon became the code name of the Australian and New Zealand Army Corps. It was however, some time before the code word came into everyday use, and at the Landing at Gallipoli many men in the divisions had not yet heard of it.

Robert Rhodes tells a similar story to Bean.

Two Australian sergeants, Little and Millington had cut a rubber stamp, with the initials A. & N. Z. A. C. at Corps headquarters, situated in Shepheard's Hotel, Cairo... When a code name was required for the Corps, a British officer, a Lt. White suggested "ANZAC". Little later claimed that he made the original suggestion to White. It was in general use by January, 1915.

 

References

C. E. W. Bean, The story of ANZAC, Official History of Australia in the war of 1914-1918, vol I and II (Sydney: Angus & Robertson, 1936)

Robert Rhodes James, Gallipoli (Sydney: Angus & Robertson, 1965)

Peter Dennis et al, Oxford companion to Australian military history (Melbourne: Oxford University Press, 1995)

Ellis Silas, Crusading at ANZAC anno domini 1915 (London: British Australian, 1916)

Fred Waite,The New Zealanders at Gallipoli (Auckland: Whitcombe and Tombs, 1921)