By the 1400s, it was a long-standing custom for the British chancellor of the exchequer to appear before Parliament once a year to present his estimates for national income and expenses. The papers containing his estimates were carried in a leather bag or portfolio. He'd place the pouch on a nearby table and then "open the budget." The budget, in this case, was the bag itself. The word derives from the Old French "bougette," a wallet or small bag. He'd open the bag and take out his papers.
A century later, "budget" came to mean only the collected contents of the bag, as in a "budget" of papers or letters. Finally, the word came to mean what we know it as today: a financial account of expenditures and revenues - whether they're in a bag or not.
Sources: 'The Encyclopedia of Word and Phrase Origins,' by Robert Hendrickson; 'Dictionary of Changes in Meaning,' by Adrian Room; Webster's Word Histories, Brewer's Dictionary of Phrase and Fable, by Ivor H. Evans; 'Why You Say It,' by Webb Garrison; The Oxford English Dictionary.