With reference to Bex (which was a proprietary name for a type of analgesic), the Australian-English phrase a cup of tea, a Bex and a good lie down, and its variants:
– denote a panacea; also, in weakened sense, a source of comfort;
– have come to be also used, often ironically, to indicate the need for a rest to settle down.

This phrase seems to have originated in A Cup of Tea, a Bex and a Good Lie Down, the name of a satiric revue by the Australian author John McKellar (1930-2010), which premiered at the Phillip Street Theatre, Sydney, on 18^th September 1965.

Amanda Laugesen, Australian National Dictionary editor, explained the following in Word Watch, published in ANU Reporter, Volume 46, No. 4:

The phrase [a cup of tea, a Bex and a good lie down] can be traced to a satiric revue of that name first performed in Sydney in 1965 at the Phillip Street Theatre.
[…]
The revue appears to have coined the phrase although variations undoubtedly had been used in Bex advertising, which often emphasised both the importance of a hot cup of tea and going to bed after taking Bex powder.

And, indeed, several newspaper advertisements for Bex associated the medication with a cup of tea (and a hot lemon drink), as well as with going to bed. For instance, the following advertisement was published in Australian newspapers from 1941 to 1945—for example in The West Australian (Perth, Western Australia) of 19^th August 1941:

“I gave him a Bex and a hot lemon drink—in the morning his cold had broken”—
writes Miss V. Rubie
265 Victoria Rd., Gladesville, N.S.W.

Neither Bex nor any other medicine is a complete treatment for colds and ’flu, but Bex does help. If you get a feverish cold or ’flu follow the treatment given below, but first read this:

Bex relieves her headaches too

Miss V. Rubie writes: “I have been ill and am now recuperating, but I feel, sometimes, headachy and off colour when I rise in the mornings. I immediately take a Bex Powder and a cup of tea, and it is surprising how much better I feel after. I have a friend living at Punchbowl and I was staying with her as she was not well. When she had a headache she also took a Bex . . . with wonderful results. Her husband had a dreadful cold coming on. I gave him a hot lemon drink and a Bex Powder, and in the morning the cold had broken and he left home next day for work.”
(Signed) V. Rubie.

How to treat Colds and ’Flu

As soon as the first symptom appears take one Bex Powder or two Bex Tablets. Before going to bed take a hot bath in a warm room, then go to bed between blankets, with a hot water bottle to the feet. Take Bex with a hot lemon drink. Diet should be liquid or light and soft, with plenty of water. Be sure to take a good saline aperient such as Kaspar Salts. For the nose and throat use antiseptic sprays and gargles. For the headache and general pains take Bex.

Try Bex with Honey for Sore Throat

A pleasant and most effective way of treating a sore throat is to mix a Bex Powder with about an eggcup full of honey and swallow the mixture slowly. Bex ingredients are antiseptic and, taken in this manner, they attack the germs at the site of the trouble.

In her review of John McKellar’s A Cup of Tea, a Bex and a Good Lie Down, published in The Bulletin (Sydney, New South Wales) of 25^th September 1965, Patricia Rolfe very clearly explained the signification of this revue’s name:

Mr McKellar, swimming over bright surfaces, is aware of depths—of “A Cup of Tea, a Bex and a Good Lie Down” as the Australian solution for any problem, social, moral or international: of the Great Australian Apathy which leads the American tourist to jump from the roof of the Chevron because nobody cares about anything here.

These are, in chronological order, the earliest occurrences of the phrase a cup of tea, a Bex and a good lie down and variants that I have found:

1-: From the caption to the following cartoon, which illustrated Patricia Rolfe’s review of John McKellar’s A Cup of Tea, a Bex and a Good Lie Down, published in The Bulletin (Sydney, New South Wales) of 25^th September 1965:
—This cartoon depicts a beggar nonchalantly reclining on his right elbow, his right hand supporting his head, his legs crossed, who says the following to the persons queuing to enter the theatre where A Cup of Tea, a Bex and a Good Lie Down is about to be performed:

“Spare us a coupla bob for a cuppa tea and a Bex.”

2-: From the following advertisement, published in The Sydney Morning Herald (Sydney, New South Wales) of 11^th December 1965:

QUEENSCLIFF.               £7250.
SWIMMING POOL, BALLROOM, HOME and INCOME

Take a cup of tea, a Bex and a good lie down—for the price above is correct, not a misprint. Amazing potential here to turn huge home with four bedrooms, living area, rumpus-room, PLUS swimming pool and ballroom into a money-spinner. Flatettes, guest house or private home for a fortunate family—anything is possible when you’ve seen this unique home right in the centre of an active, developing area.

3-: From The Bulletin (Sydney, New South Wales) of 8^th January 1966:

Trials and Troubles of Vending
New polish on the industry image
By Paul Coombes

Two major headaches are now confronting the vending machines industry—its public image and decimal currency. […]
[…]
More than “A cup of tea, a Bex and a good lie down” appears necessary to cure the headaches of “C day” and after. The Decimal Currency Board apparently does not intend to compensate operators for the cost of installing new coin and change mechanisms in the thousands of machines now operating. Vending firms have not thoroughly computed this cost, but one soft drink vendor is believed to be up for £30,000.

4-: From Brisbane Letter, published in The Australian Jewish News (Melbourne, Victoria) of 21^st October 1966:

“A cup of tea, a Bex and a good lie down.” How apt to apply to the Queensland Maccabi with their billet problems for the forthcoming Carnival.

5-: From the review, by Brian Buckley, of Profile of Australia (London: Hodder and Stoughton Ltd., 1966), by the Australian journalist and essayist Craig McGregor (born 1933), published in The Bulletin (Sydney, New South Wales) of 12^th November 1966:

A number of gratuitous political insults are thrown around. […]
Anti-Communists, in any shape or form, are not part of McGregor’s notion of a Cup of Tea, A Bex, and a Good Lie Down, and he crudely lumps them all together on the right-hand side of history.

6-: From The cup that cheers, published in The Sydney Morning Herald (Sydney, New South Wales) of 9^th December 1967:

After transplanting a human heart in Capetown this week, Professor Christian Barnard turned to his assistants and said: “I need a cup of tea.” There can be no doubt that this remark will prove the best advertisement for tea-drinking we have had for years. […]
If he had merely said, “I need a drink,” things would have been different. If he had said, “I need a cup of tea, a Bex and a good lie down,” he would have strengthened the suspicion, widely held these days, that tea-drinking is no longer as fashionable as it used to be.