The lame beggar presents an interesting question: how is it that he had the faith to grab Peter’s hand? Luke records nothing other than that Peter commanded the man to walk in the name of Jesus. There is no gospel presentation before the miracle nor is there a test of this man’s faith before Peter reaches out his hand to do the (seemingly) impossible. How, then, is it that this man had the faith to believe that Peter was neither crazy nor cruel?
Or, to flip it around, why did Peter have the confidence to extend his hand to this man in the first place? To be sure, in Peter’s speech to the crowd, he says that this man was healed by faith in the name of Jesus, but where did Peter get the confidence that this man would believe on the name of Jesus in the first place? It is not apparent to us.
Of course, we need to be clear that faith is not some preparatory work that we need to do before we receive Christ and the benefits of salvation. There is no such thing in Christianity as a preparatory season of conviction of sin before one can receive the Word. Rather, faith itself is a benefit of salvation, a gift of God. Herman Bavinck helpfully reorients us to the truth that the whole of our salvation is a gift of God when he says,
Regeneration, faith, and conversion are not preparations that occur apart from Christ and the covenant of grace nor conditions that a person has to meet in toto or in part in his or her own strength to be incorporated in that covenant. Rather, they are benefits that already flow from the covenant of grace, the mystical union, the granting of Christ’s person. The Holy Spirit, who is the author of these benefits, was acquired by Christ for his own. Hence the imputation of Christ precedes the gift of the Spirit, and regeneration, faith and conversion do not first lead us to Christ but are taken from Christ by the Holy Spirit and imparted to his own.
The point is that this man was already receiving the benefits of salvation and incorporation into the covenant of grace before he took Peter’s hand in faith. And so, from that perspective, there was no need for Peter to give this man a test before he offered what he had to give, and there is no need to wonder how this man had faith, because the answer is simple: Christ gifted him with the faith to take Peter’s hand.
Nevertheless, since faith comes by hearing, it is interesting to think about how this man had reason to believe in the power of the name of Jesus. In my opinion, it is likely that he had heard and seen what Jesus had done during his earthly ministry. After all, we’ll later find out in chapter 4 that the man is more than 40 years old. If his practice was to be at the temple asking alms, then he likely would have been aware of the name of Jesus already and even aware of the potential impacts of that name.
But whatever the prior history that this man had with the name of Jesus, at that moment Peter points him to Christ, and at that moment, the man placed his faith in the name of Jesus, grabbing Peter’s outstretched hand and experiencing the grace and goodness of God in the reversal of the effects of the fall on his body—though of course he experienced more than physical renewal.
This whole thought experiment yields a practical application: whatever past history someone has had with the Lord Jesus Christ, when you have an opportunity to evangelize, that is a moment for the person to place their faith in him. You don’t need to go looking for evidence that your witness will be effective; even faith and conversion are gifts of God.
To get a bit more practical, it’s okay to operate with a voluntary evangelistic amnesia when it comes to multiple encounters with the same person. In the course of living your ordinary life in your community, it’s okay to “forget” past encounters. Whatever prior history there might be, your job is to point that person to Jesus when given a chance; our Lord will gift as he wills.
 Herman Bavinck, Reformed Dogmatics, ed. John Bolt, trans. John Vriend, 4 vols. (Grand Rapids, Mich: Baker Academic, 2003), III.525.