Lazarus may be appropriate as clearly he was at the brink of death. But more accurately that is a Biblical reference (though arguably, being a New Testament reference may have been recorded in Greek. There are a number of Greek myths relating to rebirth, but Lazarus is not one of them. There was a mystery faith practiced at Eleusis related to Demeter who was a goddess, so I guess this wee baby's gender may determine the appropriateness of a name. Demeter, being the goddess of the harvest (or grain) was subject to an annual death and rebirth and the worship was, in more ancient times, related to human sacrifice - the year king who was feted for a year then sacrificed for the future of the crops ... (short digression, I realise).
So glad this wee lad is in safe hands now, and clearly in the nick of time...