Answer
I'm not certain of the precise origin of the phrase, but it is may be based on Ephesians 3:17, which says that Christ dwells in the hearts of believers.
It seems to me that the gist of the idea of "asking Jesus into your heart" is asking Jesus for salvation from sin, for new life, and for an intimate relationship. These are all legitimate petitions, involving repentance from sin, faith in Christ, and proper expectations of blessing. I don't see anything wrong with teaching your children to do this. I might use different words with my kids, words that are more directly to the point so that they understand them better, but that doesn't mean the common approach is bad.
That being said, there is something wrong with teaching that a person is not saved unless he or she has prayed this common prayer. The Bible does not teach us that we must come to a moment of decision, or that we must prayer a particular kind of prayer in order to be saved. It is true that most of us are saved by means of believing in our heart and confessing with out mouth (Rom. 10:9-10), but Paul does not teach that these things must be formulated in a particular way before salvation begins.
Consider your own example: You were saved when you believed (Acts 16:30-31), which probably preceded by a significant time period any moment of decision you experienced. Paul included confession in Romans 10:10 because it is a means by which God can bestow the grace of salvation (cf. "repentance unto life" in Acts 11:18). But he did not say that this is a necessary means of salvation in all cases. Writing to a church consisting primarily of adult converts to Christianity, Paul spoke from the perspective of their experience in order to encourage them to help him bring the same experience of salvation to others, particularly in Spain (Rom. 15:24,28). He was not denying that children may have a very different experience of salvation. Many theologians, including John Calvin, doubted that young children and infants were even required to have faith before they were saved, at least in the way we normally define "faith" as having formal cognitive content, awareness, assent, etc. Infant mortality is a good example case for this idea.
With your children, I would assure them that they are saved if they believe the gospel and trust God as their savior. An event in the past is useful for reminding and for making public declaration, but what really matters is what they believe and whom they trust right now.
Answer by Ra McLaughlin
Ra McLaughlin is Vice President of Finance and Administration at Third Millennium Ministries.