Order of the Phoenix ?

[balbu84]

No: HPFGUIDX 72535

The little of the book is generally explained in the book itself. Yet 
I am still wondering why the Order is called order of the Phoenix, 
and not order of the He, or of the giant sausage. I thougt it coul be 
related to the fact that it was  born again like Dumbledore's phenix 
but how could the first members have known that 15 years ago ? I 
think this is surely related to Fawkes, but the precise reason is not 
given, unless I have not carefully read. Do you have any other 
hypothesis ??
Quentin