Controlling for response bias in a forced choice memory task
Let's start with signal detection methods. The big one in memory literature is D-Prime analysis (hits versus false alarms). If the subject chooses X more times than Y, you can compare the number of times choice X is made correctly versus chosen incorrectly (as a false alarm). If that was your paradigm, then choice Y would be the correct rejection (if correct) or a miss (if incorrect).

With this in mind, you can add some degree of confidence to your paradigm. Instead of only two choices, x or y, you can give for options (1 - definitely x, 2 - probably x, 3 - probably y, 4 - definitely y). Never give the option to say, I don't know, because subjects can be unaware that he or she knows the answer, and studies show subjects can be greater than chance through unconscious recognition. Then, if you're afraid of a side bias, extracting only "definitely" answers should be a way of excluding that possibility, because if they were just guessing they would choose the "probably" answers.