You might want to check out The Grammar Room in Mehrauli. They have a a selection of board games and delish food. Also there's this Bricks Cafe in Gurgaon which is a LEGGO-themed cafe!!!
Social Street in Champa gali is a really great and affordable place for good food and playing board games like Ludo, chess, checkers, uno, Jenga etc. You should check that out.
Unlocked in 32nd Avenue in Gurgaon. (It's a very popular haunt in Ahmedabad as well). You can find all kinds of board games for free and unique ones like Settlers of Catan etc.