The clue is in the histology and anatomy rather than the misleading common name. An acoustic neuroma is really a schwannoma, a tumour of the nerve sheath cells, and it grows on the eighth nerve. The eighth nerve has a cochlear part for hearing and a vestibular part for balance, and the vestibular part splits into superior and inferior divisions.
Classic teaching pins the origin to the inferior vestibular division, where the Scarpa ganglion sits close to the internal auditory canal. That is the favoured birthplace of the tumour even though hearing loss is the loudest symptom. The hearing complaints appear because the enlarging mass leans on the neighbouring cochlear fibres.
Working through the wrong choices: the auditory division and the superior vestibular division are not the typical seed site, and the facial nerve is merely a victim of later compression as the tumour expands toward the cerebellopontine angle. Hence the answer is the inferior vestibular division.
\[\boxed{\text{Inferior vestibular division of the 8th cranial nerve}}\]