"Send Lazarus ..."
The rich man, whose name we did not know, knew Lazarus' name. Lazarus was not some random beggar; he was known by the rich man. I wonder if the Pharisees, to whom Jesus is telling this parable, had just brushed away a beggar whose name they knew.
I find it interesting that we do not hear from Lazarus in this parable. In my mind's eye (as this is not in the Bible), I see Lazarus readying himself to meet the rich man's request as Abraham stops him by lays his hand on Lazarus' shoulder. For it is Abraham who answers the rich man, noting the change in his and Lazarus' fortunes and then explaining why Lazarus cannot help him. My mental image assumes that Lazarus moved to help because it explains to me why he would be in heaven. Otherwise, the only reason given is that he was poor, and (although the parable is more about the fate of the rich man) I believe that part of Jesus' message is that there are those whom we reflexively despise who are actually true and good.