** NOTE: When I use the word he or him below, I may refer to a person in general, not necessary a man and not a woman. In proper English language, 'he' can be used as a generic term. So no political incorrectness there... 
My stand is similar to this article:
http://www.wholeperson-counseling.org/prayer/household_salvation.html.
Basically, I believe in
individual salvation. After that, God grants us that the saved person's household can be saved through him as he evangelize to his family.
Now, some people argue that God establish a household as a unit for salvation, such as Cornelius' household
(Acts 11:13-14), Lydia's household
(Acts 16:14-15), and the jailer's household
(Acts 16:31-32). But we must remember that when God works in Israel in the Old Testament (OT), he regards his people mainly as a
group (
nation, tribe, household), rather than an individual. Yet when Jesus came, he starts asking
individuals to repent and be saved. Why the difference? Personally, I believe the
Israelite culture in the OT has something to do with it. In the OT times, the head of a household (ie. the man of the house)
dictates everyone's beliefs in the house. So when Joshua decides that he is going to serve the Lord, he also decides
for his family that his entire household is going to serve the Lord. So the members of the families in the OT have much tighter relationships with each other. The same goes for OT tribes and nations. For example, when Israel was ruled by a good king, all its citizen would worship God properly. But when a bad king came, the entire nation started to commit idolatry just as the king did (except for prophets that God preserves, of course)! That tells you that a king has great influence in people's personal beliefs in the OT. Because of this culture, Cornelius household, Lydia's household, and the jailer's household can be saved through the influence of one man. (
Note: Although Lydia is a woman, her influence on her household is apparently quite strong.)
All over the New Testament (NT) of the bible, it teaches us that we each must put our faith in Christ to be saved. So if only one person in a household have such a faith, and everyone else denies God, how can these other people be saved? Also, if absolute household salvation is true, then that means immediately when a person is saved, his family will automatically be saved; And since his family includes his children, it means
their families will also be automatically saved! And then his children's children's families will automatically be saved as well, and then his children's children's children's families will be saved, and so on and so forth! In other words, after a few more generations, everyone will be saved, even though they don't believe in God!! This will definitely contradict God's Word when it says that only those who have faith and repent can be saved. God simply cannot be mocked!
Now consider these verses (
1 Cor 7:12-16):-
12 To the rest I say this (I, not the Lord): If any brother has a wife who is not a believer and she is willing to live with him, he must not divorce her.
13 And if a woman has a husband who is not a believer and he is willing to live with her, she must not divorce him.
14 For the unbelieving husband has been sanctified through his wife, and the unbelieving wife has been sanctified through her believing husband. Otherwise your children would be unclean, but as it is, they are holy.
15 But if the unbeliever leaves, let him do so. A believing man or woman is not bound in such circumstances; God has called us to live in peace.
16 How do you know, wife, whether you will save your husband? Or, how do you know, husband, whether you will save your wife?
Verse 12-14 tells how a person can influence his/her family, and this is how I see the meaning of
Household Salvation. But for those who believe in
absolute household salvation, I wonder how they are going to explain
verse 15-16, which suggests that a person may
NOT be able to save his/her household!
So, in conclusion, I believe in
individual salvation, plus God's blessing for his family that they can be saved through his (the man's/woman's) sharing with his family. Of course, his family members still have to accept Christ before they can be considered saved.
This is just my opinion, of course!

What do you guys think?