Someone I know who is a very traditional Catholic said she’d pray for me so that Mary might wrap me in her mantle. What the heck does that mean?

Many depictions of Mary show her wearing a long veil that covers her head and stretches to the ground. This is also referred to as a mantle. Many Catholics find comfort in the image of Mary holding open her mantle and wrapping it around us, protecting us as a mother would protect her children. (This symbolism is found in Eastern Orthodox Christianity as well; the protection of Mary’s veil even has its own feast day in the Orthodox liturgical calendar).
I am guessing that your friend’s comment comes from a wish for you to feel the warmth and safety of Mary’s love. It’s kind of the same thing as saying that she wants Mary to wrap her arms around you, and for you to feel the comfort that can come from a close relationship with Christ’s mother.