In my write-up today on volume 2 of John Meier's A Marginal Jew: Rethinking the Historical Jesus, I will write some about the relationship between the Parable of Lazarus and the Rich Man in Luke 16:20-31, and Jesus' resurrection of Lazarus in John 11. I will not cover every single point that Meier makes about this issue, but only what I consider to be the highlights.
Meier disagrees with the view of some scholars that the Parable of Lazarus and the Rich Man somehow influenced or gave rise to the story in John 11 about Jesus' resurrection of Lazarus. On first sight, such a view is tempting because there do appear to be similarities between the two: both discuss the resurrection of a man named Lazarus, and both hold that Lazarus' resurrection will fail to convince certain people to repent (i.e., Lazarus' brothers in the Parable, and the Jewish leaders who want to kill Jesus in John 11).
But Meier disagrees, for at least two reasons. One, there are many Jews in John 11 who do believe in Jesus on account of Jesus' resurrection of Lazarus. And, second, Meier regards the part of the Parable about Lazarus rising from the dead to be secondary----something that Luke added to the Parable----for it reflects Lukan terminology. Originally, according to Meier (if I'm understanding him correctly), the Parable was a typical stock story about the exaltation of the poor and the debasement of the rich in the afterlife. A first century C.E. parable in Egyptian demotic, for example, talks about a rich man who "is tormented in the next world while a poor man is honored" (page 826), and such a theme also appears in rabbinic literature. (But Meier also says that the part of the Parable about someone rising from the dead in order to warn people reflects a motif in Near Eastern folklore.) Meier does not think it plausible that a redactional addition to a Parable would "influence or give rise" to the Johannine story, especially since Luke was writing roughly when John was active (the end of the first century C.E.).
For Meier, what could have happened was that the Parable was influenced by a pre-Gospel version of the story in John 11. That's why the poor man is called "Lazarus" in the Parable. Originally, Meier maintains, the poor man in the story was anonymous, as are all of the characters in the parables of Luke's Gospel. But he was later called "Lazarus", presumably on account of the similarities between a later stage of the Parable and John 11.
It appears that Meier believes that there is greater likelihood that the Johannine story would influence the Parable, rather than that a small addition to the Parable would influence the Johannine story. Perhaps that's because he considers the Johannine story to have been more prominent and well-known than a mere redactional addition to the Parable.