Yes, you can achieve this result, but it has to be done through the probate process if your mother's name was on the deed when she died. This would be done by the Executor of the estate who was granted the powers to do that by the Surrogates Court, either by Letters Testamentary if she died with a will and named her children as beneficiaries, or Letters of Administration if she died without a will and the property goes to her next of kin as specified by "default" estate law principles.
The first thing to do is to examine the deed to determine how your mother owned the property with her siblings. If she owned it as "joint tenants" or "joint tenants with the rights of survivorship", then her siblings will get her share regardless of any will. Her child would get nothing.
If she owned it as "tenants in common" with her siblings, or if the deed contains no designation, then her heirs or devisees are entitled to a share either pursuant to a will or through intestacy. Either way, this should be handled through the Surrogate Court. At that point the deed should be changed.
You should consult a local attorney to examine the deed and to probate or administer your mother's estate.
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Depends on the type of deed whether joint tenancy or tenants in common. Bring it to an experienced probate lawyer in your area. Either nothing needs to be done since it may pass with the death of your mother to her siblings or you need to file a probate proceeding if there is a will or an administration if there is no will.
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Provided that what you said is correct, as to who her share should "pass on to". Keep in mind, if she left a Will then this would govern who takes. If not, then the laws of intestacy would govern. It is a good idea to "confirm" the rightful owners by virtue of a Deed, but be certain that the rightful owners are just that: "the rightful owners".
This is a complex title question that requires viewing the deed, and the estate of your mother. The short answer is that the title does not have to be changed for you to claim your interest, but it is better to do so. Multi party ownership like this is also always a problem. If you want to be bought out or have the property sold, you will need a partition case.