Mother has been appointed a guardian from the state. Can a family member get that responsibility back so they can have the say so?

You could IF: You live within the court's jurisdiction (same county). You can pass a fingerprint, background check and credit check. You are financially secure. You are able to visit the ward at least 5 times a year. You can afford to hire an attorney and finally--- all siblings are in agreement you should be the Guardian (they sign off).

Why did the state decide on guardianship? Was it given to someone associated with the nursing home? You would need to obtain a court order to change it, retain an elder law attorney, can get very expensive. You asked "get it back". Were you guardian before and it was taken away by the state? Were you notified before the appointment? Notification of interested parties, which would include you as her son, is required by law.