When the text (and the boy) states "the other egg of the phoenix", it's referring to the black egg.
Note that when it is described, the panel shows the two eggs, but only says the "Other Egg" is in the room. In then goes on to explain the difference between the two eggs shown, to contrast this "Other Egg" from the standard myth of how the phoenix is reborn. It's still just the black one that's supposed to be there (or if the white one is actually there, it is unimportant by comparison).
So the boy is thinking of the black egg. It's a mystery, a wonder, one of many in the tale, and it's a hook that keeps the memory of the story of the city alive in his mind.
As for why the black egg is focused on... (speculation reigns for the rest of this post)
Yes, the white one, the true myth of the phoenix would represent a renewal for a Baghdad that surely needs it, at least as far as the modern Baghdad the boy inhabits depicted in the comic (I do not wish to make any judgment on how accurate or inaccurate the depiction is to reality). I did find this article that tries to dive into the reasoning behind the deviation from the regular myth and what it might represent. I am not sure its conclusions are correct - the article seems to assume that nothing hatches from the egg, representing stasis and stagnation rather than rebirth and renewal, and bases the interpretation on that. That's a distinct possibility and a good opposite to the white egg, but the story leaves it ambiguous, explicitly leaves it a mystery, in both text and image (although the panel gives it a sinister spin).
A simpler interpretation, but possibly more relevant one, is that the black egg is intended to be just what it is, the hook that keeps the story in the mind of the child. The Baghdad of Dreams is the city that was sold to Morpheus, and it will live on in dreams for all time, by the terms of the bargain. So long as the story lives on, so does the city, and the black egg, the "Other Egg of the Phoenix", is just one of many mysteries and wonders (like the globe of demons, the door of fire, the merchants ridiculously convoluted tales of mystical origins for their regular fare (true? in dreams who can say...) or any number of other things that stick in my head just from memory) that hook and keep the story alive.
The boy dreams of a wondrous place, and the wondrous place lives on in his dream. It need be no more a wonder than that.