A.T. Robertson Commentary Acts 16:31

"And they said, Believe on the Lord Jesus, and thou shalt be saved, thou and thy house." — Acts 16:31 (ASV)

To be saved (ινα σωθω). Final clause with ινα 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 verse 17.

Believe on the Lord Jesus (Πιστευσον επ τον κυριον Ιησουν). This is what Peter told Cornelius (10:43). This is the heart of the matter for both the jailor and his house.