The New York Times open its archives dating back to 1851 for public perusal. Most of the articles are freely available, although you occasionally run into articles that can't be accessed without payment.
Here are some first mentions of various games and topics, by date, with a brief note about the topic covering the subject.
Avalon Hill: April 12, 1969. Adult games are a serious business.
Axis and Allies: February 18, 1984. New toys and games. Article not accessible.
Backgammon: November 15, 1852. An article exhorting people to stop playing Backgammon and write letters to their mothers and sisters.
Baseball and Cricket: November 11, 1858. A recounting of work undergone in constructing "The Central Park", including grounds for these games.
Bridge: I can't access the articles, but both contract and duplicate bridge appear to get first mentions in 1927.
Chess: December 2, 1851. Comparing a clash between the Democrats and Whigs to moves on a chess-board.
Conversation Cards: December 7, 1903. A list of game helpful for studying languages, the list also includes two French and a German game.
Cribbage: December 25, 1852. Somewhat scandalous reflections on Christmas.
Dungeons and Dragons: September 8, 1979. First mention of the game is the infamous "student goes missing at MSU" report.
Fox and Geese: June 1, 1853. A rather humorous criticism of a supposed séance. Uses the game's name as an aspersion.
December 1, 1895. Game makers hope to produce the next "hit games for Christmas" that will be the next Lotto, Tiddledy Winks, Authors, Old Maid, Snap, or Go Bang. Reversi is one new game, as are new games based on boats, horses, football, baseball, American politics, and the new bicycle craze.
Kriegspiel: July 6, 1878. Introducing the war game as played in the German army.
Magic: the Gathering: August 14, 1994. 300 million cards already sold.
Mah-Jongg: September 3, 1922. Introducing the Chinese game to American audiences, with illustrations.
Mancala: November 28, 1897. About the African game and its diffusion in Africa.
Mansion of Happiness and Muggins. June 20, 1883. Spain's King and Queen played these all night, apparently.
Pachisi: December 29, 1878. About the Indian passion for the game. People used cloth boards which they rolled up to use as turbans.
Pictionary: September 21, 1986. An article on the game makers before the game is brought to market.
Pit: November 22, 1903. Introducing the game.
Scrabble: I'm not entirely sure, but I believe 1953 is the first mention.
Settlers of Catan: November 16, 2002. Only a passing mention in an article about religious board games. Sheesh.
Snakes and Ladders: I can't access the article, but apparently first mention is November 1936. "Chutes and Ladders" in December 1937. Since the game "Chutes and Ladders" wasn't released under that name until the 1940's, I'm curious to see what the article says.
Trivial Pursuit: January 17, 1983. Can Trivial Pursuit compete for attention against the likes of video games such as Pac-man?
Whist: October 9, 1851. Recounting of the Westchester Agricultural Fair.