Phoenix takes its name from the mythical bird said to rise renewed from its own ashes, and the name is used almost entirely for that association with rebirth. It suggests resilience specifically tied to recovery -- not simply enduring hardship, but returning changed by it.
The dominant star being in the Midheaven, exalted or in its domicile and taking part of the ascendant. - Picatrix (Andalusia, ~1000.AD)