There are several places that say they provide free or low cost mammograms in New York City. Contact information can be found on the pages cited below.

According to their website, the Women's Outreach Network Mobile Mammography Program is a free service eligible to all women. You can make an appointment to be screened by calling the number on this page.

The YWCA of Brooklyn has this 2007 Mammogram Calendar which lists dates and locations for the coming months. There is a contact phone number at the bottom of the page where you can schedule an appointment.

The NYC Health and Hospitals Corporation offers free or low-cost mammograms at several sites throughout the city. See this page for locations and links to further information.

The NY State Dept. of Health has a county-by-county list of partner locations that offer free services on this page. The site refers to "eligible women," which may in some cases limit it to women over 40 (or 35), but women who have a family history of breast cancer may also be included as eligible. I would encourage you to keep calling these numbers, as the specifics of your case as expressed by you may help provide access.