What does Phoenix mean?
From Greek 'phoinix', the mythical bird that is consumed by fire and reborn from its own ashes. The fire link is through the rebirth-by-fire myth rather than a literal word for fire.
What does Theo mean?
Short form of Theodore (or feminine Theodora), from Greek theos 'God' + doron 'gift'. Carries the 'gift of God' sense of the full name; on its own the element theos means 'God'.