I think, the expression "dead of the night" means "the middle of the night", "midnight."
The American idiom "dead center" means "the exact middle". Example, "The treasure was burried in the dead center of the island".
So, I suppose, in this expression, the "dead" means the same.
I would be interested to hear the opinion of a teacher or a native speaker -- if I am correct or not.
Interested in Language