Step 1: Recall the purpose of Hinsberg's test.
Hinsberg's test distinguishes primary, secondary, and tertiary amines based on their different reactions with a specific sulfonyl chloride reagent in alkaline medium.
Step 2: Identify the reagent by its functional group.
Hinsberg's reagent is benzenesulfonyl chloride, which contains the $-SO_2Cl$ (sulfonyl chloride) group attached to a benzene ring. Its formula is $C_6H_5SO_2Cl$. The key distinction: it has $S=O$ linkages, not a simple $C=O$ as in acyl chlorides.
Step 3: Eliminate the distractors.
$C_6H_5COCl$ is benzoyl chloride (an acyl chloride, not sulfonyl). $C_6H_5CONHCH_3$ is an amide. $C_6H_5CH_2NH_2$ is benzylamine, itself an amine. Only $C_6H_5SO_2Cl$ is Hinsberg's reagent.
\[ \boxed{C_6H_5SO_2Cl} \]