When to erect the gravestone. Some people are very particular to erect the gravestone right after the end of Shivah / שבעה. Many conclude Sheloshim / שלושים with Hakamat Matzevah / הקמת מצבה — unveiling of the monument. Others hold that the monument should be erected after the twelve months of mourning, since the memory of the deceased is still vivid for the first twelve months and only after that needs to be perpetuated by erecting the monument. Waiting more than twelve months is too long to leave a grave unmarked.
Consecration of the monument. Hakamat Matzeva / הקמת מצבה — a formal dedication or consecration of the monument (sometimes referred to as unveiling, i.e., the formal removal of a veil draped over the stone) is practiced in many communities. The custom is to say Psalms, as described in the Prayers at the Graveside section, and to eulogize the deceased at the time of unveiling, recite Mourner's Kaddish / קדיש יתום and the memorial prayer. Therefore, the unveiling should not be scheduled on one of the days of communal rejoicing when eulogizing is forbidden. In order to recite Mourner's Kadish at the graveside, a Minyan / מניין — a quorum of at least ten Jewish men over the age of thirteen required for public prayer, must be present.
Pebbles. At the end of the ceremony, as always before taking leave of the deceased, the custom is to place a small stone upon the grave. The Jewish tradition of leaving a pebble on top of the tombstone probably comes from the ancient practice of Stimat HaGolel / סתימת הגולל — sealing the grave with a large gravestone to mark the grave and to prevent scavengers from disturbing the grave. In our time it shows that someone has honored the memory of the deceased by visiting the grave. Beautiful stones with the Star of David engraved on them are available for purchase here.
Making a stone grow. A tombstone is erected on the grave because, unlike living organism, a stone does not grow by itself. A stone, therefore, is symbolic of a soul that can no longer aquire merits and elevate itself after the demise of the body. However, good deeds of the living, performed in the merit of the deceased, can elevate his or her soul, just as the pebbles can make a stone bigger.