What kind of paperwork is needed for my mom to travel on airline with my daughter??

Depends on where they are traveling I would guess. Definitely a driver's license or a state issued identification card. I believe I used a birth certificate in the past to go someplace but I cannot remember the details. If you are going overseas, a passport, of course.

How old is the daughter?

Shes 8.

I would think just a birth certificate but I would give whichever airlines you plan on flying a call. They will be able to give you specifics.

The child's birth certificate will more than likely be required. You may want to go to a notary and give your mother permission to take the child as well as make decisions regarding your child. The airlines are quirky when it comes to kids who are not traveling with a parent - even if its with a grandparent.

Depending on how long they will be gone, you also may want to consider giving your mother temporary guardianship to avoid any issues.