To be saved (hina sōthō). Final clause with hina and first aorist passive subjunctive. What did he mean by “saved”? Certainly more than escape from peril about the prisoners or because of the earthquake, though these had their influences on him. Cf. way of salvation in Act 16:17.
Believe on the Lord Jesus (Pisteuson epi ton kurion Iēsoun). This is what Peter told Cornelius (Act 10:43). This is the heart of the matter for both the jailor and his house.