Maxwell did not think of the displacement current as a continuation of the conduction current, and his motivations are generally entwined with his mechanical models of ether. But the naming itself comes from the analogy between the added term and the polarization field induced by electric field in a dielectric, caused by atomic charges slightly separating and creating electric dipoles. Here is Maxwell in his own words:
"Electromotive force acting on a dielectric produces a state of polarization of its parts similar in distribution to the polarity of the particles iron under the influence of a magnet, and, like the magnetic polarization, capable of being described as a state in which every particle has poles in opposite conditions... The effect of this action on the whole dielectric mass is to produce general displacement of the electricity in a certain direction." [Maxwell, SP, 1, 496-498; quoted from Whittaker, A History of the Theories of Aether and Electricity.]
Siegel gives a detailed account of Maxwell's thought process and modern controversies surrounding it in The Origin of the Displacement Current. He explains the analogy as follows:
"Going back to the charging capacitor circuit, consider the case where there is a block of dielectric material in the space between the capacitor plates, as in figure le. In this situation, the electric field $E$ attributable to the accumulating charge on the plates would tend to polarize the molecules of the dielectric as indicated, so that the current attributable to the growing polarization would be in the proper direction to close the loop... This physical argument is only suggestive, rather than determinative, as the value of the polarization current would in all cases be less than is required to completely close the loop, and there would be no polarization in vacuo.
[...] The mathematical paradigm for the treatment of polarized media was Simeon Denis Poisson's treatment of the magnetic case; this analysis had been taken over for the electrical case by Ottaviano Mossotti, whom Maxwell cited explicitly; and standing behind Maxwell's physical interpretation of all of this was Faraday's work on dielectrics, also explicitly cited by Maxwell. The equation that Maxwell wrote for the displacement represented an amalgam of these influences, as well as the requirements of his own specific situation in "Physical lines". [Reference to On Physical Lines of Force (1861-62).]
It should be added that the motivation often cited in modern texts for adding the displacement current term, namely to reconcile the divergence-free nature of $J=\textrm{curl}\,H$ (because $\textrm{curl}$ is divergence-free) with open circuits, where electric charge is accumulating, is not supported by the historical record. Maxwell himself discusses the solenoidal current only in later works. Originally, he talked in terms of molecular vortices in the pervading magneto-electric medium of ether, and introduced a pattern of elastic strain in the neighborhood of an electric charge, a mechanical interpretation of the electrostatic field. In other words, he was thinking in terms of elastic theories of ether, popular in the 19th century, and sought a mechanical unification of electromagnetic and optical phenomena. In this context the terminology of "displacement" comes naturally. But it would be hard to bring up that context in modern etherless expositions, and invoking a purely mathematical inconsistency is far more expedient.
The magneto-electric medium was pictured as consisting of near spherical (perhaps, dodecahedral) rotating cells (molecular vortices) filled with a fluid substance, later replaced by a solid elastic material. The cell walls consisted of a layer of small, spherical particles. When the magnetic field was nonuniform, adjacent vortices rotated with slightly different angular velocities, and the small particles were carried along with the more swiftly moving vortex. Maxwell was faithful to Faraday's ideas about the "primacy of the field", and considered charges and currents to be objectionable (because unobservable) and derivative. The causality for him is reversed compared to the modern picture (initiated by Lorentz): the equations enable us to "deduce the distribution of the currents of electricity whenever we know the values of $\alpha,\beta,\gamma$, the magnetic intensities".
Maxwell calculated that the motions of the small particles are related to the spatial derivatives of the magnetic field. Identifying the small particles as electricity, and their density as the electric current, he interpreted the original Ampere's law as a relation between the (rigid) rotations of the vortices and the motions of the small particles. The role of the added term was to make the vortices deformable, to "correct the equation... of electric currents for the effect due to the elasticity of the medium", as Maxwell explained in Physical Lines.
That the substance of the vortices was at the same time assumed to be a fluid (to represent magnetic forces) and an elastic solid (to represent their rotations) was one of those contradictory properties assigned to ether that led to Lorentz's ghostly version of it, and eventually its elimination. Siegel gives detailed illustrated explanations of Maxwell's mechanical models, and many references to the secondary literature.
