First of all, let's not confuse the "Lazarus" in this parable with the "Lazarus" who appears as the brother of Mary and Martha in John's gospel. The pool of possible names that people were given in that time was somewhat limited. Just as the gospels give us multiple Marys, Johns, and Jameses, so also are there two Lazaruses in the gospels.
Secondly, the fact that the rich man was anonymous while Lazarus was named was significant, in my opinion. (In some traditions, the rich man was called "Dives", but that was only because of the word for "rich man" used by Jerome in his Latin translation of the scriptures.) For many people, the anonymity of the rich man and the naming of Lazarus would have been a reversal of what usually happened. ("Wasn't that a great party as Dives's place last night? Too bad he has to cope with that homeless bum hanging around at his gate!") In God's viewpoint, the poor and the marginalised are always regarded with dignity, including the dignity of a name.
Thirdly, playing around with "Dives" as a name for a moment, and realising the Latin nature of the name, there were these two men somewhere in Galilee: the rich man and friend of the Empire with his Latin name and the poor beggar with a definitely Hebrew name. If we were telling this story elsewhere, could we do something similar?
- An Anglo Dives and an Hispanic Lazarus in El Paso?
- Jock MacDives and Paddy O'Lazarus in Belfast?
- An Anglo Dives and an indigenous Lazarus in many Australian country towns?
- It doesn't seem that he got his theology wrong.
- It doesn't seem that he approached his faith with an insufficient level of emotional fervour.
- It doesn't even seem that he engaged in any outrageous sexual shenanigans.