This project would be primarily about the mathematical properties of the chess piece, the knight. For those who do not know much about chess, it is the piece traditionally shaped like a horse, and which moves in an 'L' shape (Two squares one direction, and then one square in a perpendicular direction). Some of the properties to look into would be the Knight's Tour problem, where the knight must touch every square of the chessboard once without repeating a square, geometric defending with knights (a common theme of what is known as an endgame, when few pieces are left), and various other quirks that show up to cause grief to mathematicians and chess players alike.
Primary Reference: Elkies, Noam D.; Stanley, Richard P. The Mathematical Knight. Math. Intelligencer 25 (2003), no. 1, 22--34.
Potential 2: "Ranking Players in Tournaments"
This project would look into how to rank players based on performances in tournaments. Topics to be looked into are how to find a winner of a tournament directly and indirectly, forming tournament groups based on strength, and how to determine relative ranking of all participants. Applications for this extend to formal ranking systems based on mathematical equations to calculate a rank that would assign a value to strength, and additionally how to pair players for matches based on said ranking.
Primary Reference: Slutzki, Giora; Volij, Oscar Ranking participants in generalized tournaments. Internat. J. Game Theory 33 (2005), no. 2, 255--270.
Potential 3: "Coupon Collector's Problem"
This project looks into the Coupon Collector's Problem in terms of a game. Each of 2 players wishes to obtain a complete set of n "coupons" by choosing randomly from the set, multiples of each coupon exist. The problem posed is how likely a player who takes the lead in the game after 2c, c some positive integer, coupons are taken is to eventually win. Another extension is to determine at what point the chances of a comeback become statistically unrealistic.
Primary Reference: Myers, Amy N.; Wilf, Herbert S. Some new aspects of the coupon collector's problem. SIAM Rev. 48 (2006), no. 3, 549--565