He is credited with the invention and popularization of the doublet (a.k.a, Word Ladder) in which one transforms one word into another, one letter at a time, with all the intermediary steps being legitimate words. Two examples:
EAST CATAdmittedly, this is really more of a puzzle or a pastime than a game.
He spent much of his professional life tutoring students in mathematics and logic. He invented a game called The Game of Logic designed to teach the fundamentals of formal logic, using a unique way of representing logic propositions with a game board and colored tokens. The game is described in a book of the same name (available from Project Gutenberg (though they seem not to have gotten the figures right) and Amazon). Ultimately though, it seems to be for one player and seems rather like a puzzle.
Carroll also thought about and devised rules for playing billiards (the British sort, played on a table without pockets) on a circular table.
In 1880, he wrote in his diary that "A game might be made of letters, to be moved about on a chess-board till they form words.". Martin Gardner (famous for his writings on recreational mathematics) took such notes from the writings of Carroll and fleshed them out into a board game in which, as I understand it, letters are placed on the first row of a chessboard and can be moved like queens to form a word in the fifth row. Meanwhile your opponent is trying to do the same thing while blocking you. (Rows 2, 3, 6, and 7 are open to everyone.) Like Scrabble, once a word has been formed, more letters are drawn from a bag, and the first row is replenished. You can buy the game here. I am thinking of playing an improvised version with a chessboard and a set of Bananagrams tiles.