What should an elderly woman do regarding missed fasting days for many years due to her lack of knowledge; should she fast the days by estimation or give charity?

Your mother must make up for the Ramadan fasts she missed after reaching puberty, as this is a debt upon her. She is not excused by her ignorance of the obligation of fasting because she lives in a Muslim land (Dar al-Islam).

In addition to making up the fasts, she must pay an expiation (kaffarah) of feeding one poor person for each day whose makeup was delayed beyond the subsequent Ramadan without a valid excuse. The amount of the expiation is 750 grams of the staple food of the country. If she is poor and unable to pay the expiation, then she only needs to make up the fasts.

The missed days should be estimated with caution. Paying expiation does not suffice for fasting as long as she is able to fast.