The Australian-English phrase to do a Melba, also to do a Dame Nellie, and variants, mean:
– to return to one’s profession after retirement;
– of a singer or other performer: to make frequent comebacks.

This phrase refers to the repeated ‘farewell’ performances given by the Australian operatic soprano Dame Nellie Melba ¹ (born Helen Porter Mitchell – 1861-1931)—as explained in this extract from her biography, published in the Australian Dictionary of Biography:

Melba returned to England and on 8 June 1926 gave her farewell performance at Covent Garden. […]
Melba now began a series of farewell appearances that, in ‘doing a Melba’, was to enrich the language as well as bolster her self-esteem. As early as October 1924 she had announced her Australian farewell to grand opera, but her last operatic performances […] occurred […] in Sydney on 7 August and in Melbourne on 27 September 1928. Two months later in Geelong she gave her last Australian concert.

¹ Melba, the stage name adopted by Helen Porter Mitchell, is a shortening of Melbourne, the name of the capital of Victoria, Australia, where the singer was born and made her concert debut. She gained worldwide fame for her coloratura singing, and was created a Dame of the British Empire in 1918. It was in her honour that the French chef Auguste Escoffier (1846-1935) coined pêche Melba, the name of a dessert of ice cream and peaches served with raspberry sauce or syrup, in Le Guide culinaire: Aide-mémoire de cuisine pratique (Paris: L’Art Culinaire, 1903).
—Cf. also notes on the phrase ‘sing ’em muck’.

Portrait of Dame Nellie Melba (1890)—image: State Library of South Australia:

These are the earliest occurrences of the phrase that I have found, in chronological order:

1-: From Town Talk, by David McNicoll, published in the Daily Telegraph and Daily News (Sydney, New South Wales) of Tuesday 19^th February 1946:

McKell is apparently going to do a Dame Nellie Melba act.
He says “Farewell”; the A.L.P. executive says, “No. Not yet. You’re too valuable!”

2-: From Truth (Brisbane, Queensland) of Sunday 8^th January 1950—about the Australian cricketer William Alfred Brown (1912-2008):

Brown has now hit 501 runs (nine times out) in this season’s Shield matches. His form (to be continued in this match, with the Victorian game to come) may make Bill do a Melba reverse and change his mind in regard to his announced retirement.

3-: From George Hart’s Radio News, published in The Sun (Sydney, New South Wales) of Tuesday 20^th May 1952:

Looks like the Colgate Show is going to do a “Melba” on us.

4-: From Truth (Brisbane, Queensland) of Sunday 28^th September 1952:

Jockey Peter Morgan […] did a Madam Melba act lately in announcing his retirement, and then returning almost immediately.

5-: From Woman’s World, published in The Herald (Melbourne, Victoria) of Tuesday 23^rd March 1954:

A woman, who is retiring on March 31 after 25 years in a public position, and determined, she says, “not to do a Melba act,” will be given a farewell party by more than 4000 people.
She is Dr Lucy Meredith Bryce, honorary director of the Red Cross Blood Transfusion Service since it was started in 1929.

6-: From The Argus (Melbourne, Victoria) of Monday 22^nd November 1954:

Front page news abroad
Trouper Betty ‘did a Melba’

Las Vegas, Sunday
Betty Hutton ², who retired 11 days ago, agreed to replace entertainer Sammy Davis, jnr. ³, at a nightclub on Friday after Davis was injured seriously in a car crash.

² Betty Hutton (born Elizabeth June Thornburg – 1921-2007) was a U.S. actress, dancer and singer.
³ Samuel George Davis Jr. (1925-1990) was a U.S. singer, dancer and actor; he lost his left eye in an automobile accident on Friday 19^th November 1954.