If Luke 16:19-31 is a parable, then it is the only parable where Jesus gave one of the characters a name. Commenters often argue that the character of Lazarus was well known to Jesus' audience, and that they were more likely to receive this story as a literal telling of what happened to Lazarus in the afterlife.
Is there anywhere in the Bible that might give us hints as to who this Lazarus was, and why Jesus gave him a specific name?
Lazarus of Bethany
Some argue that Lazarus of Bethany here was intended. Some say that Jesus was near Jerusalem (John 10:22, Luke 13:22) when he told the parable of the rich man and Lazarus, and during the 4 days that Lazarus of Bethany was dead. Both Lazaruses died of a sickness, and the resurrection of either Lazarus could not persuade the Jewish leaders of the truth about Jesus the resurrection.
While this is possible, there is another character that seems much more compelling.
Eliezer of Damascus
Jesus likely spoke this parable, not in Greek, but in the common oral tongue of the Jews. In Hebrew, Lazarus is rendered Eliezer, and literally means "God has helped."
Genesis 15:2 tells us that Abraham's chief steward was a gentile named Eliezer. This is possibly the same servant that Abraham commissions to find a wife for his son in Genesis 24:2. If so, then Abraham likely sent this trusted servant on other errands. Such an errand to test the hospitality of Sodom is recorded in Jewish writings, and cited by J. Duncan M. Derrett in his book Law in the New Testament:
Those that live as the Rich Man, heedless of warnings, will have the fate of Sodom: for the story of Lot and Sodom is a story for perpetual information, not an old tale to be read and forgotten (cf. 2 Pe 2:6, Jude 7). Isaiah 1:10-31 commences, "Hear the word of the Lord, ye rulers of Sodom..." Ezekiel 6:46-59 contains the words "Sodom thy sister hath not done... as thou hast done... Behold, this was the iniquity of thy sister Sodom; pride, fulness of bread, and prosperous ease was in her and in her daughter; neither did she strengthen the hand of the poor and needy. And they were haughty..., therefore I took them away..."
The lawlessness of the Sodomites lay, according to Jewish tradition, not so much in their sodomy as in their inhospitableness. They were, of course, governed by the law applicable to the descendants of Noah, but their country was a kind of topsy-turvy land in which all rights and justice were subverted. The rabbis make merry over the strange rules that obtained there. With Jews, hospitality to strangers and wayfarers was a cardinal virtue; with the Sodomites it was a vice. "Eliezer, servant of Abraham, complained of being attacked: 'Give them a fee for bleeding you,' said the judge! The beds of Sodom were notorious; they were truly Procrustean, for the rule was that if the visitor was too long they lopped off his feet, and if he was too short they stretched him. The rights of the traveller were completely abrogated. A girl gave bread to a poor man secretly: she was cruelly punished. When a poor man came to a place citizens each gave a single coin, on which the donors wrote their names, but no food: when he died each took back his own coin. It was a rule that whoever invited guests to a banquet should lose his upper garment. 'Eliezer, servant of Abraham, went to test their hospitality. People asked him, 'Who invited you?' He named one after another of those present; they ran away in turn for fear of losing their clothing, till he was left alone–and so could partake of the feat. The Babylonian Talmud solemnly repeats these tales, complete with biblical citations and a comment by 'Rab'. The whole is appended to a Mishnaic discussion whether the men of Sodom would (i) partake in the world to come, and (ii) stand in the day of judgment (for trial). The final opinion (with which Jesus evidently agrees) is that notwithstanding the wickedness of the Sodomites they shall have the benefit of trial on the day of judgment; but it was evident that it was hotly debated. This is the background to our story [of the rich man and Lazarus].
Eliezer and Abraham test the hospitality of Sodom and Gomorrah and the surrouding towns in the Old Testament. Their destruction is cited as an example of those who suffer punishment of eternal fire (Jude 7, 2 Peter 2:6). Ezekiel 16:48-50 warns Jerusalem that she is far more wicked than Sodom and Gomorrah. The punishment of Jerusalem will be the same as the punishment of Sodom and Gomorrah if they do not repent. In Matthew 11:20-24, Jesus warns Bethsaida, Chorazin, and Capernaum that it will be more tolerable for Sodom in the day of Judgment than it will be for them.
Sodom and Gomorrah are a type of hellfire. It was a literal example in the Old Testament that awaits a prophetic fulfillment at the judgment. Interestingly enough, Sodom was situated in a plane that was well watered, and resembled the paradise of Eden (Genesis 13:10). Abraham lived not too far away from this location. The burning of Sodom and Eliezer in Abraham's bosom may likely be the setting of our parable.
Could it be possible that in this parable, Jesus uses this same imagery of Sodom and Gomorrah to tell us to be hospitable to the poor brought to us, as he does in Matthew 25:31-46? If so, our treatment of the rich man in this parable should be the same as our treatment of Sodom and Gomorrah: literal characters whose destruction on earth are a symbolic warning of the judgment at the second coming.