What is the basis for naming the p-orbitals with reference to the Cartesian coordinate system as x, y, and z?
The choice of axes is arbitrary in this case (isolated atom with no external influence).
You could choose about any set of co-ordinate axes, and that wouldn't really affect the behavior of any of the electrons there ;)
However, in the presence of external influences, such as a magnetic field, it's customary to take the positive axis in the direction of the field as the positive z axis. Though ultimately, the choice of notation is left to you.

EDIT: Since you seem to have an issue here, I'll mention this as well:
Quoting from chemguide.co.uk
A)  $\mathrm{d_{xy}}$,  $\mathrm{d_{yz}}$,  $\mathrm{d_{zx}}$

The names tell you that these orbitals lie in the x-y plane, the x-z plane and the y-z plane respectively.

Each orbital has four lobes. Notice that each of the lobes is pointing between two of the axes - not along them.

B) $\mathrm{d_{x^2 - y^2}}$, $\mathrm{d_{z^2}}$

Although these two orbitals look totally different, what they have in common is that their lobes point along the various axes. That's different from the first three where the lobes pointed in between the axes
The $\mathrm{d_{x^2 - y^2}}$ orbital looks exactly like the first group - apart, of course, from the fact that the lobes are pointing along the x and y axes, not between them.



The $\mathrm{d_{z^2}}$ orbital looks like a p orbital wearing a collar! The main lobes point along the z axis.