Answer
The full name is copper(II) iodide.
Work Step by Step
The charge on Cu must be 2+ for the compound to be charge-neutral with two I- anions. The name for $CuI_{2}$ is the name of the cation, copper, followed by the charge of the cation in parentheses (II) and the base name of the anion, iod, with the ending - ide. The full name is copper(II) iodide.