Answer
The full name is mercury(II) bromide.
Work Step by Step
The charge on Hg must be 2+ for the compound to be charge-neutral with two Br- anions. The name for $HgBr_{2}$ is the name of the cation, mercury, followed by the charge of the cation in parentheses (II) and the base name of the anion, brom, with the ending - ide. The full name is mercury(II) bromide.