Guardianship alone will not make the child eligible for Social Security dependents benefits based on your mother's work record. In order for a (non-disabled) grandchild to be eligible for Social Security dependents benefits -- which would be paid to your mother as part of her retirement benefits (which she can claim as early as age 62) -- the following factors must be present:
- the grandchild must live with and actually be cared for by the worker on whose record the payment are to be made (in this case, your mother), and
- the grandchild's parents must be deceased or disabled.
The other way the grandchild could qualify for Social Security dependents benefits, based on your mother's work record, is if your mother formally adopts the child.
If the child's parents are alive but potentially might be considered disabled -- for emotional or physical reasons -- it may be worth it for your mother to explore the possibility of obtaining dependents benefits for her grandchild on that basis. She should make an appointment with her local Social Security office and bring with her to the appointment any information she can gather about the child's parents, along with her legal guardianship papers for the child.