Usually the executor named in a will will start the probate process, if there is a will.
Any of the children can start the probate process and apply to be the administrator if there is no will.
IF there is no will, one of the children you apply as administrator of the estate. If there is a will, someone was named as executor, then that person can start the process. Be sure to take accounting of any part of the estate, including possessions, that have been sold or tossed. You should contact an attorney in this matter.
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This answer is not intended to provide you with specific legal advice regarding your situation, or to create any attorney-client relationship.
As the first two answers indicate, if a deceased person had a Will which names an Executor for the estate, then the first available nominated Executor generally has the responsibility to either refuse to serve or proceed with determining whether it is necessary to probate the Will. If a deceased person did not have a Will, then an Administrator may be appointed.
If the deceased person was a Georgia resident (this is not clear from your question), then if there is a Will, state law requires that the person who has or comes into possession of the original Will file it with the probate court of the county in which the deceased had her primary residence, even if no probate will be needed. This would mean that, if the daughter-in-law in your situation claims to have a Will from the deceased which names her as Executor and gives her the deceased's assets, she has to produce the original and file it with the probate court at some point, even if she claims no probate is needed. If the daughter-in-law is not claiming that the deceased person had a Will, then she may well be simply acting without authority.
If a Georgia resident dies without a Will, the person's heirs are generally given the ability to select an Administrator. The "heirs" are the person's closest living relatives, as determined by Georgia law. In the case you describe, assuming there was no surviving spouse, the dececdent's heirs would be the living children, plus any living grandchildren by the deceased child (I assume there was only one predeceased child plus the 4 living children), and plus the living children of any predeceased grandchild. The daughter-in-law would not be an heir unless the decedent had legally adopted her prior to death. If any of the heirs are minors, or if all of the heirs cannot agree on an Administrator, it can get complicated very fast.
One task is to determine whether any assets actually became part of the decedent's estate. If the decedent owned the house or cars, you need to be able to see exactly how those assets were held. You also need to be able to determine how any bank or brokerage accounts were held, and whether any beneficiary designations, "payable on death," or "transfer on death" designations apply to the accounts. If the decedent owned any asset solely in her name, with no beneficiary or other designation, then it is likely that at least that asset became part of her estate and someone will need to be appointed as Executor or Administrator to deal with it. However, depending on what the decedent (or the daughter-in-law) did during the decedent's lifetime, it may be that no significant assets became part of the decedent's estate. For example, the house and cars could have been transferred out of the decedent's name (or into her name and the daughter-in-law's name as joint tenants), and any accounts could have been with the daughter-in-law as a joint owner or as a designated beneficiary.
The decedent's heirs should strongly consider consulting their own attorneys to determine what their options are. If no one has asked the daughter-in-law directly about the situation, someone should also try to do so. She may be willing to talk, or she may not be. However, either response should at least provide some indication of what the heirs may be facing.
Good luck to your family with this matter.
Your fact pattern did not state whether or not the mother had a will. If their mother had a last will and testament then normally the person who is named in the will would offer it for probate thus starting the probate process. However there are certain instances where the person who is named as the executor cannot or refuses to serve. In such a case then you would again look to the will to see if there is an alternate named executor and if not then a person who would be interested in the estate (i.e. an heir) would file the will and offer it for probate. Of course, the law prefers those who are named in the will to serve and if not then blood relatives. If the mother died without a will then an heir or interested person such as a creditor could file a petition for administration seeking to be appointed administrator of the estate. This is much more complicated than merely offering a will for probate because of the notice that would have to be published. After the period for the notice has run you would hold a hearing before the Probate Court judge to determine if your petition will be granted or not. In your fact pattern because the mother with surviving children they are the heirs. I realize that the daughter-in-law has alienated the heirs however they cannot sleep on their rights. What this means is that they have to determine if their mother had a will. In order to determine if their mother had a will they could contact the attorneys in the area to determine if their mother had contacted any one of them to prepare a will and if so request that the attorney file the will directly to the Probate Court. If they cannot find a will then they will need to proceed with an administration. In either case I would strongly suggest that they find a competent probate attorney to handle this matter.
Disclaimer: I am licensed in GA and FL. Please note that this answer does not constitute legal advice, and you cannot rely on this answer as legal advice. I suggest that you consult with another attorney of your choosing so that he or she may properly evaluate your question through a comprehensive consultation and review of all the facts and documents at issue. My answer does not create an attorney-client relationship between us.