Often, the family of the deceased will have an unveiling service for the presentation of the matzevah.* What is the origin of this custom?
*Source: Goldin, Hyman. Hamadrikh. Hebrew Publishing Company: New York City, 1939. Print.
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Nitei Gavriel - Aveilus 2 67:3 says that after searching in all the Sefarim of many different Kehilos, Chevra Kadishas, and Aveilus, he could not find a source for an unveiling. He mentions on the bottom that the Steipler was once discussing this with his son Reb Chaim Kanievsky, and he said that it is most likely not of Jewish origin.
The newer Madrikh, by Rabbi Bulka, says the point is for the family to get together to make sure the tombstone is properly in-place (hence the Hebrew term, "hakamat matzeiva"); he then says the "unveiling" practice, whereby the stone is first covered by a cloth and then given a dramatic reveal, "has no basis in Jewish practice whatsoever."
(Me: yet it seems everyone does it...)
The concept of the unveiling stems from the need to clearly mark the gravesite. Removing the cloth from stone symbolizes erecting the tombstone. Making sure that the gravesite is marked properly is a form of honoring the deceased.
In this sense, it would appear that removing the cloth is a similar gesture to placing a stone on the gravesite (a reminder that the family is there, and an ancient way of marking gravesites).