Legal Guardianship for the Children of a Family Member

Becoming a legal guardian in Germany is difficult (lots of legal training required) and not a short-term option.
Convincing the authorities (specifically, the "Jugendamt") to place a child under a legal guardian is even more difficult: You'd basically have to show that the biological mother is permanently incapable of caring for the child. This is usually a first step to open the child up for adoption.
I believe that both are not what you really want.
If you just want to have the child stay with you and take care of him/her while the mother is studying, there is no guardianship required - just a mutual consent between the mother and you. (For some formal things like moving or school registration, she'd still need to sign forms or give her consent, but this is manageable - if need be even by post!)

As Beppi said, no legal guardianship required. You may need a power of attorney from the mother for emergency situations.

Much more interesting will be how to get a visa under these circumstances.

Will the mother live far away from you?

Is there a university near you that she could apply to or is she set on a specific institution?

EDIT: Actually, there is a possible spanner in the works. Where is the father of the child?

ALKB wrote:

Actually, there is a possible spanner in the works. Where is the father of the child?

Very good point: In Germany, unless you have sole custody (as declared by a court of family registry), the father has to give consent to all major decisions in the child's life. The mother alone cannot register him/her for school, schedule a medical treatment (except emergency), move to another address or travel abroad.
This is handled much more lenient in many other countries: I have had several discussions (also on this forum) with single parents who thought, just by not being married (which makes no difference here), not having contact with the other parent and not knowing where he/she is, they have the full say in everything regarding the child. Not in Germany!
Even if the father has passed away, your relative would have to prove this to the German authorities with an official death certificate.