Ab initio effective potentials
First off, I would like to note that you wrongly interpreted the very first equation. Below I quote the relevant part of the paper:


  In the Hartree-Fock (HF) approximation for, say, the $\ce{Na}$ atom,
  the ground-state wave function has the form
  
  $$
\Psi = \mathcal{A} (\Phi_{\mathrm{core}} \phi_\nu X) \, , \tag{1}
$$
  
  where $\Phi_{\mathrm{core}}$ is a product of (ten) spatial orbitals
  very similar to the orbitals of $\ce{Na+}$ and $\phi_\nu$ is the
  valence orbital (the one removed in ionizing to $\ce{Na+}$). ($X$ is an
  appropriate product of spin functions.)


So, $\Psi$ is not just some function of spin orbitals designated by a weird letter as you said, it is, as usual, their antisymmetrized product, where I bet the calligraphic A symbol ($\mathcal{A}$) as usual stands for the antisymmetrizer. The only small difference is that small calligraphic a is used in the text rather then much more common capital one.



Now to you actual question. Both core and valence orbitals are thought in the derivation as the solutions of the Hartree-Fock equations, which have a very well-known feature: they are defined to within a unitary transformation (see, for instance, here). So, orbitals which are solutions of the Hartree-Fock equations are not unique: there are infinite different sets of orbitals that minimize the Hartree-Fock energy and you are free to choose any of these set for any further work. This is, essentially, what is said in the paper:


  It will be important in our later analysis to note that the valence
  orbital in (1) is not determined uniquely by the variational
  principle. One can modify $\phi_\nu$ by mixing in an arbitrary amount
  of any core orbital (doubly occupied. in $\Phi_{\mathrm{core}}$)
  without changing the energy.


Just note that this is true not only for the valence orbital: all orbitals (i.e. core as well) are not uniquely determined by the variational principle, as I mentioned above.

Now, if the goal is to have a unique valence orbital, then it can be achieved by restricting it to be orthogonal to core orbitals. In that case when transforming orbitals you can not mix any amount of any core orbital into the valence one since it will immediately make it non-orthogonal with this particular core orbital. And since mixing in core orbitals is the only way you can change the valence one, orthogonality of the valence orbital implies its uniqueness. 


  In order to obtain unique solutions for $\phi_\nu$, one generally
  requires that $\phi_\nu$ be orthogonal to the other occupied orbitals.


And it is important to note (as it is done in the next sentence of the paper) that :


  Although this orthogonality restriction has the desired consequence of
  leading to a prescription for a unique valence orbital, there is no
  reason to consider the orthogonal valence orbital $\phi_\nu$, to have
  a special significance over any other combination of $\phi_\nu$, with
  the various core.




Note finally, that due to the orthogonality restriction you can not also mix any amount of the valence orbital into the core ones, but you can mix core orbitals with themselves anyway you like since it won't do any harm for their orthogonality with the valence orbital. Thus, while the valence orbital is uniquely defined by the orthogonality requirement, the core orbitals are still defined to within a unitary transformation.