Get a lawyer, this is not appropriate for self help. And specifically, hire an attorney who has considerable experience handling probate (estate) litigation in the county in which the public administrator was appointed. You need an expert that has the right background, and contacts/experience, in the same geographic area. If your mother is the legal guardian of the ward (her daughter) it would seem that the court would be considerate of her wish, but does she have the investment backround to deal with this? If the funeral home bequethed to the handicapped daughter is sold and the proceeds invested, it might be simpler for your mother to then petition the court to take over. A local probate litigation expert will be able to help you assess what your mother would have to demonstrate to the court to take over. Bear in mind, the court probably appointed the public administrator for a reason. You and your lawyer will have to convince the court that your mother is now a better option. Also, this is not likely to be inexpensive.

As another option, if your mother has specific concerns that the administrator is not addressing she should meet with him/her and communicate her concerns (in writting). It might be a lot cheaper and simpler to encourage the administrator to take actions your mother wishes to benefit her daughter, than to change the administrator. Consider this option first. Also, this option may be the first step to replacing the administrator. If your mother has reasonable requests of the administrator that are in the best interests of the ward, and the administrator is not responsive, and your mother documents the fact that the administrator did not respond, this might be the foundation for a suit to remove him/her as administrator.

Coming full circle, consult a lawyer. This is not a job you should attempt on your own.