In Judaism is there any meaning behind the age 65 or for being a great-grandmother?
I am constructing some words to say to my mother, and I’d like them to relate to Judaism.

There is only significance in the fact that she is old and wise (old age is considered a step in holiness. It isn’t a bad thing.) – experienced in seeing G-d through the world and her life’s experience. She is considered a link in the eternal Jewish chain.

You may find this Mishna in Pirkei Avot chapter 5:25 interesting:

He (Rabbi Yehuda ben Taima) used to say, at five [one should begin the study of] Scriptures; at ten, Mishna; at thirteen [one becomes obligated in] the commandments; at fifteen [the study of] Talmud; at eighteen the wedding canopy; at twenty to pursue; at thirty strength; at forty understanding; at fifty counsel; at sixty old age; at seventy fullness of years; at eighty spiritual strength; at ninety bending over; at one hundred it is as if he has died and passed on from the world.

Rabbi Meir Goldberg