Jeff,
I cannot locate any records for your mother or her mother in the DAR. You mentioned that your mother was a Lifetime member, this would lead me to believe that she would most likely have several pins and a ribbon (or ribbons) that she would have worn on the left side of her chest for DAR events. If you have any of those, or even photos, I would love to view them and perhaps we can determine if we are on the right path.
I did locate information on Mary Alvis Draper for you. DAR does have a "Patriot" file for her. Several women used her as their Patriot line. However, due to changes in requirements, any future applicants must now use the current requirements.
I did find that there is an article for Mary Draper and her service that you can look up: revolutionary-war.net/mary-draper/
Looking up a chapter for DAR is not simple. I would need to know what State you mother lived in at that time.
If you would like to contact me via email, feel free. (tammiecochran@yahoo.com) just put DAR in the message line please.