I suppose I used to think it was a sort of a spare promise in the Bible. But the day came when I realised that that promise, with its "you and your household," contains a divine principle that you can see throughout the word of God.
Consider. The Bible tells how mankind became so desperately wicked that God decided to put an end to it. But Noah found grace in His sight. God told him to build an ark. When it was time to enter the ark, God said to Noah (Gen 7:1): "Come into the ark, you and all your household." The Bible says that Noah was a righteous man. It doesn't say anything about his family being righteous - but because of Noah, eight people were saved.
Abraham was God's friend. God made a covenant to bless him. The sign of the covenant was circumcision. Circumcision was not just for Abraham, but for his sons - not just for Isaac, but for Ishmael - and not just for his sons, but for his servants (Genesis 17). In short, for Abraham and all his household.
When it was time for the last of the 10 plagues in Egypt, the Israelites were instructed to slay a lamb and put its blood on the lintel and doorposts of their homes. That night, the firstborn in every household in Egypt would die, but the firstborn in each house covered with the blood would be saved. They were instructed to slay a lamb not for each individual, but for each household (Ex 12:3. 7, 13).
Jericho was under condemnation. Everything in Jericho was to be destroyed. But Rahab sought the Lord. She was told that she and everyone with her in her house would be saved. It doesn't say anything about her household having sought the Lord. But because of Rahab, Rahab and all her household were preserved. You can see the promise in Josh 2:18, 19 and its fulfilment in Josh 6:17, 22, 23, 25.
Ah, you say, but all that was in the Old Testament. Come with me to the New Testament. Zacchaeus was a chief tax collector, and a money-grabber. When Jesus went to his house, something happened. "I give half of my goods to the poor," said Zacchaeus, "and if I have taken anything from anyone by false accusation, I restore fourfold."
"Today," replied Jesus, "salvation has come to this house" (Luke 19:9). What a strange thing to say. He didn't say that salvation had come to the man, but to the house.
Cornelius, a Roman centurion, saw a vision of an angel. "Send men to Joppa," said the angel, "and call for Simon whose surname is Peter, who will tell you words by which you and all your household will be saved" (Acts 11:13, 14).
When Paul and his colleagues preached to Lydia in Philippi, Lydia and her household were baptised (Acts 16:14, 15). When the jailor in the jail at Philippi asked "What must I do to be saved?" Paul and Silas told him: "Believe on the Lord Jesus Christ, and you will be saved, you and your household" (Acts 16:31 - 34). And so it happened.
There are other examples of households being converted, like the household of Crispus (Acts 18:8) and the household of Stephanas (1 Cor 1:16).
When Joshua was an old man, he called all Israel together and reminded them of all that God had done for them. It was time for them to choose whether they would serve the God of Israel or whether they would serve other foreign gods. "But as for me and my house," said Joshua, "we will serve the Lord" (Josh 24:15).