There are many different customs as to when one should unveil the monument over a grave. Not only is it permissible to wait until after the first year has passed, many authorities encourage this. The reasons given for waiting until after 12 months since the deceased was buried are:
1) The monument itself is a symbol of importance, which is seen as inappropriate during the mourning period while there is still pain.
2) One purpose of the monument is so that the deceased will not be forgotten from one’s heart, but during the first year the memory of the departed is still vivid, as many mourning practices ensure, so we are not concerned about the possibility of forgetting them so soon.
However, many argue that the purpose of the monument is for the deceased and their honor, and it is proper to give them this respect as soon as possible. For this, and other reasons, most authorities encourage the monument to be erected, at least partially, right after one gets up from sitting shiva, and then completed, if necessary, before the 30th day. Some even allow one to leave their house during shiva in order to prepare the monument because they view creating it as soon as possible to be very important.
That said, we must acknowledge that there are many customs as to when to construct the monument over a grave, ranging from right away, to one year (which in my experience is the most common custom), to only after one year has passed. Therefore, one should follow whatever the prevailing custom is in their community, but if there is a compelling reason that one needs to wait until after a year, they may certainly do so.
It has become customary for mourners to gather at the cemetery for a ceremonial "unveiling" of the headstone of a loved one who has died. This is a powerful ritual moment to bring together friends and relatives to reflect on the legacy of the deceased and to pay honor to their memory.
In general, the custom is for the unveiling to happen at approximately the one year mark following a death (see Lamm, "The Jewish Way in Death and Mourning," Klein, "A Guide to Jewish Practice," etc). This is a natural time for such a gathering, as it can mark the conclusion of the formal period of mourning.
However, since unveiling is a purely a minhag (custom), unknown to either the Talmud or the Codes of Jewish Law, there is no halakha (law) to when or how it must be done. While the custom has become to do it within the first year, if for some reason the ceremony is delayed, there is no reason why it cannot take place at a later time. If family and friends find it meaningful and comforting to gather and recall the life of someone who has died, there is no reason to deny them that opportunity for healing.
According to Isaac Klein, in his book A Guide to Jewish Religious Practice (Jewish Theological Seminary of America, 1979):
The usual practice is to erect the tombstone no sooner than one year after the death, since to do it earlier would be an indication that the memory of the deceased is fading and artificial means are required to revive it…In Western European countries and in America, it has become the custom to “consecrate” the tombstone with a service. Since in America the tombstone is covered with a cloth, which is removed by the family during the service, the ritual has been called the “unveiling.”
The unveiling usually takes place twelve months after the death. While there is no traditional basis for this service, except for custom of visiting the grave on the day of Yahrzeit, it is now an accepted and meaningful practice. It offers additional opportunity for the officiant to pay tribute to the deceased as well as to speak to the living about the meaning of life and death. Participants should take care to insure that the unveiling does not become a social event.
Alfred J. Kolatch, in his book Inside Judaism (Jonathan David Publishers, 2006), writes the following:
Although monuments have been erected over graves for many centuries, the custom of conducting a special unveiling ceremony was instituted toward the end of the nineteenth century in England and later in the United States to formalize and dignify the erection of the monument. The British use the term “tombstone consecration,” whereas Americans have adopted the term “unveiling.”
There is no religious obligation that an unveiling be held, and it is not necessary that a rabbi officiate. Anyone able to recite the selected psalms and deliver a eulogy, if one is desired, may conduct the ceremony.
Although an unveiling may be held at any time after the monument has been erected, most families wait about one year so as to give the earth a chance to settle.
Based on the above explanations—namely, that unveiling is a custom rather than halakhic (decreed by normative Jewish law), and a relatively recent custom at that—it would seem to me that an unveiling may be done at almost any time when the surviving family members deem it an appropriate time, and one in which everyone whom they would like to be in attendance can attend.
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