Here's how to break down the name:

* Bi: Represents the element Bismuth.

* I: Represents the element Iodine.

* 3: Indicates that there are three Iodine atoms for every one Bismuth atom.

* (III): This Roman numeral indicates the oxidation state of Bismuth, which is +3 in this compound.

Therefore, the full name of BiI3 is Bismuth(III) iodide.