Step 1: Identify all functional groups in the compound.
The benzene ring compound contains: $-CN$ (cyano group), $-Br$ (bromo), $-OH$ (hydroxy).
Step 2: Determine the principal characteristic group by IUPAC priority.
IUPAC suffix priority (highest first): $-COOH > -CN > -OH > -NH_2 > -X$. Here, $-CN$ has highest priority; parent compound = benzonitrile, with C1 bearing $-CN$.
Step 3: Assign positions to substituents.
With C1 at $-CN$: from the compound structure, $-Br$ is at C2 and $-OH$ is at C3. Numbering chosen gives lowest locants.
Step 4: Name substituents with their positions.
Position 2: bromo. Position 3: hydroxy.
Step 5: Construct the full IUPAC name.
List substituent prefixes alphabetically: bromo (B) before hydroxy (H). Full IUPAC name: 2-Bromo-3-hydroxybenzonitrile.
Step 6: State the final answer.
\[ \boxed{2\text{-Bromo-}3\text{-hydroxybenzonitrile}} \]