Is it permissible to benefit from a ring gifted by a young girl to her mother with the mother's consent? And is a mother entitled to dispose of her minor children's property?

A minor's gift is not valid, whether their guardian permits it or not, because they are restricted from disposing of their property to protect their wealth. It is also not permissible for a guardian to donate anything from a minor's wealth, due to the Almighty's saying: ﴿And do not approach the orphan’s property except in a way that is best﴾. The guardian is liable for whatever they donate from the minor’s wealth. Therefore, it is not permissible for your sister to donate the ring; rather, she must keep it safe for her daughter.