Try to use colors that inspire calm feelings. Pastels, especially blue and yellow, are great choices. Steer away from fiery colors like red or orange as they can be hostile.
Ask everyone in the family their favorite colors! Get paint chips (they're free!) in these colors and have a meeting to discuss the colored chips. Choose something everyone likes!
You want a family area to be bright, especially if you have kids. Yellow is a good color, blends in well and provides a feeling of warmth and cosiness.