The ancient Babylonians chose to organize their calendar around 7 days of the week, and each day corresponds to one of the seven "planets" visible to the naked eye. [Planets is in quotations because two--the Sun and the Moon--are technically not planets (they're luminaries).]
Monday (lundi in French) is ruled by the Moon. "Monday's child is fair of face" might refer to the sensitivity and youthfulness associated with the Moon.
Tuesday (mardi in French) is ruled by Mars. "Tuesday's child is full of grace" might be more aptly interpreted as full of energy and movement.
Wednesday (mercredi in French) is ruled by Mercury. "Wednesday's child is full of woe" might be interpreted as a person who worries or thinks too much.
Thursday (jeudi in French) is ruled by Jupiter. "Thursday's child has far to go" might thus be interpreted as a person who will travel and learn.
Friday (vendredi in French) is ruled by Venus. "Friday's child is loving and giving" might refer to the Venus nature, that is, a person who is loving and receptive to love.
Saturday (samedi in French) is ruled by Saturn. "Saturday's child works hard for a living" thus might refer to the Saturnine nature which is rather serious and responsible.
Sunday (dimanche in French) is ruled by the Sun. "But the child born on the Sabbath Day is fair and wise and good and gay" might refer to the magnanimous Sun nature.
This is also the case in Japanese (and most likely Chinese as well):
Monday (Getsuyoobi) - "Moon day."
Tuesday (Kayoobi) - "Fire day." Mars (Kasei) is "Fire planet"
Wednesday (Suiyoobi) - "Water day." Mercury (Suisei) is "Water planet"
Thursday (Mokuyoobi) - "Tree day." Jupiter (Mokusei) is "Tree planet"
Friday (Kinyoobi) - "Gold day." Venus (Kinsei) is "Gold planet"
Saturday (Doyoobi) - "Earth day." Saturn (Dosei) is "Earth planet"
Sunday (Nichiyoobi) - "Sun day."
I have to say though, I hope we don't end up with a day named after Pluto, as "Dark King day" does sound rather scary!!