Although this may be the case through some kind of accident, I have never heard of any school offering more aid money to female applicants than male applicants.

Is this perhaps some strange application of affirmative action? I've heard of schools offering more positive application decisions/financial aid to people of minorities or difficult economic backgrounds, but never because of gender.

Affirmative action in the United States can target women as beneficiaries, but it happens very rarely. Studies that have aimed to clarify the effect of affirmative action on colleges' admitted students have ignored gender. Usually, race or ethicity is the main thing affected by affirmative action. So, it can make a difference in your financial aid, but it's likely to be very small, even negligible.