E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
The statement which the Chancellor of the Exchequer lays before the House of Commons every session, respecting the national income and expenditure, taxes and salaries. The word is the old French bougette, a bag, and the present use arose from the custom of bringing to the House the papers pertaining to these matters in a leather bag, and laying them on the table. Hence, to open the budget or bag, i.e. to take the papers from the bag and submit them to the House.
A budget of news is a bagful of news, a large stock of news.
Cry Budget. A watchword or shibboleth. Thus Slender says to Shallow
We have a nay-word how to know one another. I come to her in white and cry mum; she cries budget; and by that we know one another.Shakespeare: Merry Wives of Windsor, v. 2.