I don't know whether to count REM's Man in the Moon: "Let's play Twister, let's play Risk, ..." The song is about Andy Kaufman.
An entire album devoted to Cheapass Games by the Beatnick Turtles. They also did the intro for Geekspeak.
Chess, the Musical. John Greshak has an entire page devoted to more songs about chess. So does Bill Wall.
Mousetrap on Pac-Man Fever, by Jerry Bruckner and Gary Garcia. You can listen to a snippet.
Stratego, in German? I need someone to translate.
Scrabble: scroll down for lyrics (some naughty words) by Otis Lee Crenshaw here. Or listen to most of a song about Scrabble by Emily Kaitz here.
Backgammon: Backgammon By the Bay has a page of songs to listen to while playing backgammon, including a link to a composition called Backgammon.
Monopoly: Listen to a snippet from Monopoly, the theme song (from the short lived game show). Other snippets or lyrics: Urge-Overkill, Richard Berman (click to start the song), Rivermaya, Donsen Park, Martinets.
I'm leaving out a number of instrumental-only songs titled after games, such as "The Game of Draughts", "Chinese Checkers", and so on.
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