1839, a trade name for a type of woolen fabric:

MICHAEL NOWAK, alias John Mazurkiewiez, was indicted for stealing on the 15th of April 2 ¼ yards of woollen cloth, called tweed, value 12s., and 2 ¼ yards of woollen cloth, called doe skin, value 17s., the goods of George Priestley Heap. [London Central Criminal Court minutes of evidence from 1839]

This apparently developed from the "Tweed Fishing or Travelling Trousers" advertised in numerous publications from 1834-1838 by the clothing house of Doudney & Son, 49 Lombard Street.

So celebrated has amateur rod-fishing in the Tweed become, that the proper costume of the sportsman has now become an object of speculation among the London tailors, one of whom advertises among other articles of dress "Tweed Fishing Trousers." The anglers who have so long established their head-quarters at Kelso, for the purpose of enjoying the amusement of salmon fishing in the Tweed, have had excellent sport lately : some of the most skilful having caught five or six salmon a day, weighing from six to fourteen pounds each. [New Sporting Magazine, June 1837]

Thus ultimately named for the River Tweed in Scotland. The place name has not been explained, and it is perhaps pre-Celtic and non-Indo-European.