A parent is forbidden from also being a godparent, but other relatives, including grandparents, are allowed. My pastor has discouraged people from automatically asking aunts and uncles to be godparents with the explanation that those people will be in the child's life anyway; godparents are an opportunity to bring additional faithful adults into the child's life. Perhaps that explanation would make your mother feel better about not being chosen.
Here are the relevant sections of canon law:
Can. 874 §1. To be permitted to take on the function of sponsor a person must:
1/ be designated by the one to be baptized, by the parents or the person who takes their place, or in their absence by the pastor or minister and have the aptitude and intention of fulfilling this function;
2/ have completed the sixteenth year of age, unless the diocesan bishop has established another age, or the pastor or minister has granted an exception for a just cause;
3/ be a Catholic who has been confirmed and has already received the most holy sacrament of the Eucharist and who leads a life of faith in keeping with the function to be taken on;
4/ not be bound by any canonical penalty legitimately imposed or declared;
5/ not be the father or mother of the one to be baptized.