Rebbe,
The gemara says that the woman can write: "I have received a maneh," thereby lowering her kesuba's value by half if she so chooses. No matter what her motivation is it's still a sheker. How do we approve someone signing on to a lie?
B'kavod,
Sam Kosofsky
It is not a lie. Just as a person can forgo a loan, a woman can forgo an obligation of payment that the Torah says her husband has to her because of her Kesuvah. Saying "I have received a Maneh" is basically saying that I consider it as if I have received sufficient payment for this Maneh.
Another way of explaining this would be similar to a girl who owes her mother money, and the mother says that she will forgo the loan since she has been a good girl in school this week. It is not a lie for her to say she received a Maneh, as she received compensation equal to her to a Maneh from her daughter. A woman who is marrying a man and forgoes a Maneh probably deems herself lucky enough to marry this man that it is as if she received a Maneh.
All the best,
Yaakov Montrose