Peter's healing of a beggar at the temple gate, then, is a way of saying the Peter deserves to be heard; not surprisingly, this incident is followed by a Petrine sermon, one remarkably similar to the one that he preaches in Acts 2. It is impossible to know how much of Acts is compilation versus composition on Luke's part. The healing/sermon combo in Acts 3 could be part of an oral tradition or an earlier source from which Luke is drawing. He could be combining elements from different places. Given the similarities between the sermons, this could be a variant of the Acts 2 message, which Luke has paired with a healing story as a narrative device. Or it could have simply happened, and gotten preserved in oral tradition or reconstructed by Luke on the basis of eyewitness accounts. It's not like we will ever know exactly where these stories came from.
Peter is, in effect, the new Jesus. Though his message is undeniably about Jesus, it is also a continuation of Jesus's own message, and his miracle-working is also very reminiscent, serving a similar purpose in the narrative. Jesus healed people; Peter heals people. Jesus preached repentance in light of the coming Kingdom; Peter preaches repentance in light of the coming Kingdom. Jesus is not here, but he's returning; Peter is here, to tell you so.
Peter preaches with and in the authority of Jesus, which means there is a kind of circularity here: Peter preaches about Jesus in the name and authority of Jesus, whom we are expected to believe is the Messiah because Peter tells us so in the name of Jesus. And why should we believe Peter? Because he just healed a guy, 'nuff said. This is the only thing that points outside Peter's argument. It's similar to citing a source or quoting a recognized authority.
Peter's (or Luke's) appeal to authority is no more or less bogus than our appeals to authority. It's different, to be sure, reflecting the different world in which Luke lives. Sometimes a healing story serves as a kind of parable in and of itself, like John 9 for example. Other times, like this one, the story really just tells us something about the healer or miracle-worker in question. The point is not, exactly, that Jesus or Peter could heal people. The thing-in-itself is not really the thing at all. This is why, even though we've all done it, it's pointless to wonder why Jesus just didn't heal everyone and be done with it. Broad theological speculations notwithstanding, such questions simply miss the point.
As such, then, the healing of the lame man in Acts 3 is -- I don't want to say it's not important -- not important. It doesn't tell us much except by way of giving us a concrete example of Peter's wonder-working power. Enter lame man, Peter heals lame man, Peter makes another stump speech for Jesus. End of story. The lame man is like a Red Shirt -- the expendable unknown crew member in a Star Trek episode -- except instead of getting killed, he gets healed, which is a lot happier but still doesn't do anything for his acting career.
There is, however, an interesting tidbit almost buried in here. Peter and John were headed to the temple at the hour of prayer. Why, exactly, were they doing this? Let me suggest two reasons: One, they were headed to the temple to proselytize. This is plausible, especially since that's exactly what they end up doing. (Or at least Peter does. For all that I admire about Luke's writing skill, he lacks facility with multi-character dialog. John is pretty much just furniture in this story.) I don't like this explanation as well as the other one, which is that they were headed to the temple to pray because they were Jews in Jerusalem.
It doesn't matter if Jay and Silent Bob actually did this or if it's just something Luke created to set up Peter's sermon. The upshot is that the idea of Penn and Teller heading to the temple at 3 in the afternoon apparently didn't strike anyone as odd, and it seems reasonable to assume that the early Jesus followers remained temple-attending Jews until they became a liability to Jewish community as a whole, or until the Temple was destroyed. Jesus' disciples were not spinning off a religious alternative to Judaism so much as they were cultivating a political alternative to Roman imperialism.
It's true that Luke's narrative looks ahead to the inclusion of the Gentiles, but Luke's is not some simplistic replacement theology, or supercessionism. For now, regardless of Luke's agenda, and perhaps even irrespective of mine, we can make the observation that Christians in Jerusalem were observant Jews, or at least, wherever Luke is from, that idea is plausible enough to be the backdrop for a healing narrative.
Back to the story. I've heard sermons that try to make a lot out of the healed man's response -- the jumping and leaping and praising God and all that. Usually, the point is that we should emulate his enthusiasm. Sometimes the point is that this man just got his legs healed; we, receiving the very salvation of our souls, should be even happier. But really, what person, having just been healed of a crippling disease, wouldn't jump and leap simply because they could? And what can you do after you've been saved? Die and go to heaven?