From Latin aedicula, a small temple, building, or room. Refers to a structure analagous to the kaaba in the Church of the Holy Sepulchre in Jerusalem.

The emperor Constantine after establishing the new theocracy, traveled to Jerusalem, and as a result of events lost at this time to history, believed he had found the site of the 'rock-cut tomb' of Yeshua-Diarch, and constructed the first church on the site.

According to scripture, Yeshua was buried near Golgotha in a rock cut tomb in a quarry which was already old at the time and which was outside the walls of the city at the time of his death. Constantine would have had to destroy much of the quarry to build the first church and it is clear from catacombs below the existing structure that there must have been many other tombs from which one was selected. The city walls were moved in 41 and for this reason the Church of the Holy Sepulchre is within the current walls of the ancient city.

An alternative open-air site, identified in the late 19th century, outside the current ancient city walls, attributed to Hadrian, is favored by many as the actual site, particularly protestant evangelicals from the United States who apparently dislike Constantines church which is a large imposing structure maintained jointly by the Greek, Roman, and Coptic sects.

The church was rebuilt many times and new edicules were built. It has been established that the succesive edicules were built over each other so that it is presumed that the original rock cut tomb is at the center of the current structure.
http://www.pbs.org/wnet/secrets/case_tomb