What does Acts 3:6 mean?

When faced with a lame beggar, Peter offers something better than money: healing. He provides that healing through the name of Jesus. "Name" is from the Greek root word onoma. It's not just an identifier. "Name" encompasses the essence of the person: in this case, Jesus' character, sovereignty, authority, power, deity and identity as God. Peter doesn't use "Jesus Christ of Nazareth" as a magical spell. He has faith in the Person of Jesus. He has the power of the Holy Spirit (Acts 1:8; 2:1–4). He has authority from Jesus to heal if it is for Jesus' purposes (John 16:23). So he has the confidence that Jesus will heal and honor his words (John 14:13–14).

Some may have faith that Jesus can grant what they ask, but that doesn't mean they can superficially ask "using" Jesus' name and get what they want. James warns, "You ask and do not receive, because you ask wrongly, to spend it on your passions" (James 4:3). Others may have good exterior motives, but no faith, like the Sons of Sceva who tried to exorcise a demon "by the Jesus whom Paul proclaims" and got beaten up for their trouble (Acts 19:11–16). Part of praying in Jesus' name means submitting to His purposes. John makes this clear when he says, "if we ask anything according to his will he hears us" (1 John 5:14). Since we don't always know Jesus' will, it's enough to ask for what we think we need and trust that He will take care of what we really need.

Why didn't Peter have any money? This isn't the first time he's been found without coins. When faced with tax collectors, neither he nor Jesus had any cash. Instead, Jesus told Peter to go fishing and that the fish would have the coins they needed (Matthew 17:24–27). When Jesus and His disciples travelled, Judas held their money (John 12:6). In the very early days of the church in Jerusalem, the believers held their property in common; those who had need were provided for by those who had extra (Acts 2:42–47). Carrying currency, at this point, is simply not a priority for Peter.