What is the origin of "rag" meaning newspaper?
The origin of this use of the word goes back to the seventeenth century. The OED’s entry for it comes under the category rag used in ‘Senses relating to something compared to a torn piece of cloth’. It is quite possible that early newspapers bore just such a resemblance.
To supplement Barrie England and MετάEd's answers: Etymonline reports that the pejorative term rag, which is used to express or suggest a newspaper's worthlessness, dates back to the 18th century.
rag (n.) scrap of cloth, early 14c., probably from Old Norse rögg "shaggy tuft," earlier raggw-, or possibly from Old Danish rag, or a back-formation from ragged, […] As an insulting term for "newspaper, magazine" it dates from 1734
The first British newspapers were actually called gazettes after the Italian gazzetta, which in turn is said to derive from the Italian word for magpie (la gazza ladra). And one of the first and oldest surviving British “newspapers” is called The London Gazette. It was first printed on 7 November 1665 under the name of The Oxford Gazette.
The first general-interest magazine was The Gentleman's Magazine, first published in 1731, in London. Whether this magazine or subsequent periodicals inspired the epithet rag (documented three years later), is not known. It might have been an ironic term used by its own readers, or a gratuitous insult by its opponents, there's no way of knowing for sure.
Poor paper quality not writing
From an article published by the same magazine, in 1823, the following extract supports the theory that the term rag was used to refer to the main component of a newspaper, and perhaps, initially, to the substandard quality of the paper used in printing these journals etc. during the 18th and 19th century.
It is notorious that the great mass of printing papers are now made of cotton rags; and that to produce a better colour, the pulp undergoes a chemical process, which materially injures its durability —EDIT
A reader writes:
Allow me to call the attention of your readers to the present state of that wretched compound called Paper. Every printer will corroborate my testimony † and I am only astonished that the interesting question has been so long neglected and forgotten. It is a duty however of the most imperative description our beautiful Religion our Literature our Science all are threatened.
I have seen letters of a recent date already become a carte blanche. One letter, which I forwarded by post, fell to pieces by the way, and I have noticed more than once a description of writing-paper, that being bent, snapped like a bit of watch-spring. I have in my possession a large copy of the Bible printed at Oxford, 1816 (never used), and issued by the British and Foreign Bible Society, crumbling literally into dust.
[…]
The causes of destruction are twofold : the materiel and the mode of bleaching the rags.
The use of cotton rags was very happily superseded by those of linen, yet I fear some manufacturers are not very scrupulous in the selection.
The application of quicklime to the rags, once prevalent in France, but very properly subsequently interdicted was a serious evil for it actually decomposed the material. Are we entirely guiltless? Such a process must needs disorganize the fibre. […][Emphasis in bold mine]
The Gentleman's Magazine From July to December, 1823
Due to the high demand of white paper in the 18th century, the processes employed at the time was responsible for the reduced longevity of paper, as confirmed by BAPH (British Association of Printing Historians)
Increasing demands for more paper during the late eighteenth and early nineteenth centuries led to shortages of the rags needed to produce the paper. Part of the problem was that no satisfactory method of bleaching pulp had yet been devised, and so only white rags could be used to produce white paper. Chlorine bleaching was being used by the end of the eighteenth century, but excessive use produced papers that were of poor quality and deteriorated quickly. By 1800 up to 24 million lb of rags were being used annually, to produce 10,000 tons of paper in England and Wales, and 1000 tons in Scotland, the home market being supplemented by imports, mainly from the continent. Experiments in using other materials, such as sawdust, rye straw, cabbage stumps and spruce wood had been conducted in 1765 by Jacob Christian Schäffer.