In Ghost in the Shell, the word ghost is colloquial slang for an individual's consciousness. In the manga's futuristic society, science has redefined the ghost as the thing that differentiates a human being from a biological robot. Regardless of how much biological material is replaced with electronic or mechanical substitutes, as long as an individual retains their ghost, they retain their humanity and individuality.
The concept of the ghost was borrowed by Masamune Shirow from an essay on structuralism, "The Ghost in the Machine" by Arthur Koestler. The title The Ghost in the Machine itself was originally used by a Gilbert Ryle to mock the paradox of conventional Cartesoam dualism and dualism in general. Koestler, like Ryle, denies Cartesian dualism and locates the origin of human mind in the physical condition of the brain. He argues that the human brain has grown and built upon earlier, more primitive brain structures, the "ghost in the machine", which at times overpower higher logical functions, and are responsible for hate, anger and other such destructive impulses. Shirow denies dualism similarly in his work, but defines the "ghost" more broadly, not only as a physical trait, but as a phase or phenomenon that appears in a system at a certain level of complexity. The brain itself is only part of the whole neural network; if, for example, an organ is removed from a body, the autonomic nerve of the organ and consequently its "ghost" will vanish unless the stimulus of the existence of the organ is perfectly re-produced by a mechanical substitution (this isn't necessarily true, think of pain in phantom limbs). This can be compared, by analogy, to a person with innate hearing disability being unable to understand the concept of "hearing" unless taught.
Ghost-dubbing, or duplicating a ghost, is an impossibility in the Ghost in the Shell universe. When performed, as a cheap AI substitute in Innocence and earlier in the manga, the original wears off.
In Ghost in the Shell, Kusanagi completely reproduces the stimulus of all of her organs in order to maintain her "ghost". If a technical error arises during the transfer of a "ghost" from one body to another, the transfer normally results in failure, since the "ghost" tends to deteriorate due to either the difference of system at the material level or the deficiency of the transferring protocol. The Puppet Master manages not to deteriorate its "ghost" when merging with Kusanagi because his system is the body of information itself, thereby avoiding a deterioration due to the deficiency at material level.
The Greeks had a similar paradox, called the Ship of Theseus. Hegel's concept of Geist may also be related.