The 2e product Aurora's Whole Realms Catalogue is a unique accessory that is written in the form of an in-game catalogue of a company selling all kinds of items throughout the Realms. It has a section titled Diversions, which covers toys (balls, toy arrows and bows, marbles, dolls, etc.) as well as games. There we read about "four universal games": chess, draughts, dice and talis cards. In addition, there are three more games that are sold: fighting dolls (foot-high dolls controlled by sticks), "Old Men's Bones" (a game of pick-up sticks), and "Table Dice" (backgammon).
For a more recent (November 8, 2018) semi-official answer, you can also have a look at the following series of tweets from Ed Greenwood, the original creator of the setting:
... Off the top of my head, we have “play games” like jacks, shove-skittles, tag, Toss the Dagger, the Xorvintaal game played by dragons for social standing by accumulating points derived from real-world accomplishments (Whispers of Venom by Richard Lee Byers, and references in Erin Evans’ novels). There are also dice (gambling) games like thabort and Traitors’ Heads. There are a few trackboard-and-dice games like shirestone (played by halflings) and Wheel-of-Spells (Cormyr: A Novel), High Dragon, Smashcastle, Strikedragon/Battles, Swords, Swords and Shields, and Wandsiir. As well as card games that use the ... Talis deck
Then there are the board games, from the familiar to us of Earth draughts/checkers, chess/lanceboard, and go (both referenced in the Red Knight entry in 2e Powers & Pantheons), to: sava (a chess-like drow boardgame with many complex side options), chethlachance, fiveknights, lancers and lions, and more. I even snuck a reference to the Lords of Waterdeep boardgame into DEATH MASKS. And then there are card "teaching games" (nobles' Who'sRelatedToWho).