This project would be primarily about the mathematical properties of the chess piece, the knight. 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, 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. Applications for this extend to formal ranking systems based on mathematical equations to calculate rank, and additionally to 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 is to eventually win.
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