Name meanings for characters:

[theredqveen]

Prudence/Prue:
    meaning 'caution' or 'discretion'. prudence was classically considered to be one of the four cardinal virtues, as it's associated with good judgement, practical wisdom, and foresight. 

Piper: 
    meaning divine communication, influence over children and animals. 

Phoebe: 
    meaning 'radiant', or 'bright'. phoebe was the goddess of earth's moon, forests, wild animals, and hunting.

Paige: 
    meaning 'helper', 'assistant'. has connections to the spiritual element of earth which represents stability, being grounded, and nurturing.

Additional characters:

Penelope/Penny: meaning 'weaver', long suffering but loyal
Patricia/Patty: nobility, wisdom. lady of the house
Andrew/Andy: courage, leadership. 'manly', or warrior-like
Leo: meaning voice, or command. disciplined and noble  
Cole: victory, spirituality and inner strength. dark, swarthy

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What's interesting, is given the meanings of the name of the Core Four, I think they should actually be named differently:

• Prue should be Phoebe - the protector of innocents and women in particular, goddess of hunting (Prue loved to hunt demons)
• Piper should be Paige - stability, groundedness. Piper is the most helpful of the sisters, she would help Grams with cooking and was the mediator between her siblings for years
• Phoebe should be Prue - foresight meaning premonitions, and good judgement could later equate to empathy (knowning how somebody is feeling, revealing their inner self etc). the flipside, meaning caution could relate to Phoebe's rebellious teenage years
• Paige should be Piper - divine communcation, or allowing the elders to talk through her as she is half whitelighter, and her work in social services relates to the influence over children, and as for 'animals'... Paige did have the closest relationship with the more magical creatures of the Charmed universe