I can but speculate the possible reasons that Schrodinger would have chosen "Psi".
Firstly, Algebra has many differing letters that are common to equations (For instance C<SUP>2</SUP>)
Rather that label everything A, B, C. Schrodinger might have decided that using the Greek Alphabet would make more sense (Of course he could have drawn circles, squares, triangles, but it would have looked odd)
I would guess if that was the reason he chose it, it would be down to the fact that he was tutoring.
Another point was that because he was tutoring and Quantum Mechanics in it's day was in Theoretics for a majority of it's discussion as equipment hadn't been devised to explorer it. This would mean that you would have to use a "Conscious thought" to construct meaning to a Hypothetical experiment.
This would pretty much give reason to the reason why he Picked "Psi" as the wave formation he was trying to forwards as theory, other mechanics didn't believe existed. So it would be a mark of Conscious thought about it.
One last one I made up for a laugh, He used it just to annoy Heisenberg.