Firstly, The primary purpose of writing this is to be an encouragement for believers to believe and to give Scriptural aids in prayer, that The Kingdom of God may come to the Households of believers in particular in this case, with increased righteousness, peace and joy in the Holy Spirit, Romans 14:17; and this writer is therefore looking for reasons to believe, not doubt. The Westminster Shorter Catechisms states, in Question. 99, "What rule hath God given for our direction in prayer? Answer. The whole word of God is of use to direct us in prayer; but the special rule of direction is that form of prayer which Christ taught his disciples, commonly called the Lord's prayer." This is an example of how the Word of God can direct us in praying for our households, since a Biblical, Historical and Modern Day precedent has been established, the chief cornerstone being the Biblical precedent.
Also, general principles don't have to work in every instance, for some, any or all the reasons given below, because an exception, by way of definition, does not invalidate a general principle. We see this with speed limits and people, ambulances and law enforcement heading to hospitals or to the scene of some occurrence. It is the same with Scripture, and the instances in itself are too numerous to list... But when you have promises as clear as the the Scriptures stated in the original post (Acts 16:31), and historical and modern day experiences coinciding, I would say we have to remember that the blessings in Deuteronomy are given to GENERATIONS of those that love Him (Deut 28). The Lord's very description of His own name in Ex 34:6-7 says he shows mercy to GENERATIONS of those that fear Him. We can't forget Paul's whole point in Romans 9-11 is that is isn't up to him who wills or runs, but God who shows mercy. That's the Sovereign Father we are coming alongside as friends with Christ, asking Him to save our family members on the basis of His Will which is His Word.
It's the exception not the rule that The child of an unbeliever believes, and that is why God gave Ezekiel the exception phrases in Ezekiel 18, that the children aren't punished for their parents sin but their own sin, and that the soul that sins will die. It is the sin of the son of the unbeliever that the unbelieving son of the unbeliever will die from, as well as the unbelieving son of the believer. But with parables as strong as "The Prodigal Son" and scripture and revival examples as strong as the ones above, why not pray according to that? I will concede the biblical Pater-familias had a strong sway over his household, but I contend the modern father's life, belief, prayers and intercessions, when using the authority of Jesus' Name, are still just as powerful and effective, James 5:16, they avail much. It is God's will that households come into the kingdom, and over the years, mine have pretty much one by one. I believe, that as Jesus said, of all the Father has given me I will loose not one. I don't look at their behavior when I pray to God, but his promises.
Have you ever heard the story of George Mueller who prayed for many people to come to salvation and one seemed not to have repented? That man repented at George Mueller's funeral! It is God's permissive will to "take you one from a city and two from a family, And I will bring you (them) to Zion," Jer 3:14, however God's perfect will is to for us to pray for our unsaved family members, ask His mercy on their lives, pray God brings witnesses into their paths and understand "Salvation Belongs To Our God" (Jonah, Revelation, John 6, Romans 9-11). Put our unsaved family members on our prayer lists, others' prayer lists, & web sites. Confess the Word over them, God wills none would perish, Peter says, that all would be saved and come to the knowledge of the truth, Paul says. That's enough proof for me. There is boat loads more, but I have people for which to pray. Connect with the sovereign God who sits in the heavens and does whatever he pleases (Ps 115:3). He pleases to save souls. Here are the testimonies and quotes from George Mueller and Hudson Taylor, from an exceptional link for inspiration and testimonies to intercession, below.
One day George Mueller began praying for five of his friends. After many months, one of them came to the Lord. Ten years later, two others were converted. It took 25 years before the fourth man was saved. Mueller persevered in prayer until his death for the fifth friend, and throughout those 52 years he never gave up hoping that he would accept Christ! His faith was rewarded, for soon after Mueller's funeral the last one was saved.
On persevering prayer: "I look at a stone cutter hammering away at a rock a hundred times without so much as a crack showing in it. Yet at the 101st blow it splits in two. I know it was not the one blow that did it, but all that had gone before."
"Eighteen-year-old Hudson Taylor wandered into his father's library and read a gospel tract. He couldn't shake off its message. Finally, falling to his knees, he accepted Christ as his Savior. Later, his mother, who had been away, returned home. When Hudson told her the good news, she said, "I already know. Ten days ago, the very date on which you tell me you read that tract, I spent the entire afternoon in prayer for you until the Lord assured me that my wayward son had been brought into the fold." Our Daily Bread, July 19, 1989. [Hudson Taylor (1832-1905) was a famous missionary in China. He was founder of the China Inland Mission which, at his death, included 205 mission stations with over 800 missionaries, and 125,000 Chinese Christians. He spent 51 years in China].
http://www.tentmaker.org/Quotes/prayerquotes4.htm