Step 1: Structure of benzylamine.
Benzylamine consists of a phenyl group attached to a methanamine group ($\mathrm{C_6H_5-CH_2-NH_2}$).
Step 2: Applying IUPAC nomenclature.
The parent chain is methanamine, and a phenyl group is attached to it, giving the name phenylmethanamine.
Step 3: Elimination of other options.
Phenylethanamine has two carbon atoms in the side chain, benzenamine refers to aniline, and N-methylbenzenamine is a different substituted amine.
Step 4: Conclusion.
Hence, the correct IUPAC name is (B) phenylmethanamine.