Would I be able to donate some of my hair follicles to my mom to get a hair transplant in case her own hair couldn't do the job?

You cannot donate your hair to your mom because the grafts will reject.  Only way to donor hair to another person is to your genetically identical twin.

Unfortunately the answer is no. Hair transplant is actually a tissue transplant, and would require anti-tissue rejection medication if donated from one person to a different person.

Hello,
The term for this is hair allograft and it cannot be performed because hair contains individualized cellular markers that will be rejected by the immune system system of the new host. In order to achieve this, your mom would have to go under immunosuppressant drugs - the side effects would greatly outweigh any advantage that a hair transplant may bring. Research this site to find alternative solutions to solve your mom’s hair-loss.

I'm afraid not. Only identical twins can donate hair follicles to each other. Have your mother visit an expert at hair loss, and there may be other alternatives.
Thank you for your question, and best of luck to your mother.

Other than between identical twins, hair follicles cannot be safely transplanted from one person to another. 

Unfortunately you can not donate your hair for your mom. The transplanted hair will not take. The donor would need to have the exact same DNA. That was a very nice gesture on your part.

Hair is an organ so when it is transplanted to another person it undergoes rejection unless the two people are identical twins so the answer is that it can not be done successfully