Neo - New.
Nym - Name, moniker, etc (search TIL for "protonym"). Remember that in a magical setting a name isn't inert, it has a power association with the thing named.
Biosis - life, living, etc.
There are different ways you can link those terms together, but the meaning is basically the same any way you do it. And that meaning fits the context perfectly:
"Enter as Lord Dagon has written: come slow and bring four keys. Know that then you are royalty, a new breed of destroyer, whose garden shall flood with flowers known and unknown, as it was in the mythic dawn. Thus shall you return your first primal wail and yet come out different. It shall this time be neonymbiosis, master akin to Master, whose Mother is miasma."
Or if it isn't obvious from that, think about a neophyte advancing into the higher orders of a religious or mystic group, or think of the symbolism of baptism in Christianity (not intending to start a discussion of religion here, it's an example). Now consider what this means in a world where myth is fact.