Sgr is short for Sagitarrius, a constellation. If you mark off the boundary of Sagitarrius, Sgr A* would be in that region. The black hole itself does not have the name, but rather the radio source associated with it. (conjecture alert)I'm not sure where the * comes from, but I believe the A is associated with the fact that Sgr A is a super nova remnant. Supernova are typically named "SN [year] [alpha]" where [year] is the year discovered and [alpha] is an alphabetical character in order of discovery. A supernova remant wouldnt have a year associated, so i'm guessing it's given a constellation and then alphabetical marking.
Typically, astronomers have a mix of common names and many systematic naming schemes, but they get obsoleted when better instruments find darker objects. There was once a catalog of nebula called Messier with about 130 objects (M1,M2,..). We still use these designations, but suddely they found thousands more of these objects and had to rename using the New General Catalog, (NGC1,NGC2,NGC496). Even further the IC goes into many thousands.
For constellations, stars in each constellation are usually named alphabetically with greek chars in terms of their apparent brightness. Alpha Centauri is the brightest star, for example, in Centarus. Then there are common names like Polaris, the north star, whch is alpha ursa minoris.