Interesting question - I believe the Greek letter naming scheme is  chronological in nature - the polymorph discovered first would be alpha, the next would be beta, and so on.  Now, I've not read that this is so, but having looked at and studied the literature on many different crystals, that is how it seems to me.

In fact, there is no single rule. The most typical convention is ordering of letters according to the temperature intervals of stability. Sometimes α-phase is the one stable at the lowest temperature (like Fe, Sn or quartz), sometimes it is the one stable at high temperature (like WO3). Sn and WO3 are examples of the substances for which α-phase is not the first one chronologically (β‐Sn and γ-WO3 are stable at the room temperature).

When high pressure or metastable phases are included in such list, they usually go after phases that are stable at the ambient pressure. In this case chronological order is usual.

Sometimes even more complicated cases exist. For example, stable phases of Bi2O3 are α- (low temperature) and δ‐ (high temperature), while β‐ and γ-Bi2O3 are metastable. And for Al2O3 β‐phase does not exist at all (the material called initially β‐Al2O3 appeared to be NaAl11O17.

You can compare all four forms of this hydrated oxide. The crystallographic results lead your to certain suggestions about the difference :) Good luck!

Thank you for your valuable answers, but the actual reason behind the naming of the phases is still not very clear. If somebody know the exact logic, please respond.