Leaving stones or pebbles on a grave is an ancient Jewish tradition, but its origins are unclear. It is not a commandment, rather a custom or tradition. Over time, many different interpretations have been offered for this practice.
Common explanations include:
A warning to Jewish priests known as Kohanim
During the times of the Temple in Jerusalem, Jews began marking graves with piles of rocks as a way of warning passing kohanim that they should stay back. The Jewish priests (kohanim) became ritually impure if they came within four feet of a corpse.
To keep the souls in this world
The Talmud, also referred to as the Shas, mentions that after a person dies, his or her soul continues to dwell in the grave where he or she was buried. Jews believed that placing the stones on a grave would keep the soul down in this world. Some people find comfort in this. Another interpretation suggests that the stones will keep demons and golems from getting into the graves.
Stones last longer than flowers
Flowers, though beautiful, will eventually die. A stone can symbolize the 
permanence of memory and will not die.
A Hebrew Pun
Rabbi Simkha Weintraub, rabbinic director of the New York Jewish Healing Center once said: “The Hebrew word for ‘pebble’ is tz’ror – and it happens that this Hebrew word also means ‘bond.’ When we pray, the memorial El Maleh Rahamim prayer (and at other times), we ask that the deceased be ‘bound up in the bond of life’ – tz’ror haHayyim. By placing the stone, we show that we have been there, and that the individual’s memory continues to live on in and through us.”
Special care is usually taken when selecting a stone to put on a loved one’s grave. It may be from a place of meaning to the deceased, or simply an interesting or attractive rock. Because there is no commandment behind placing a stone, this action serves as an opportunity for you to create your own, meaningful ritual.