Half Shekel was Atonement for Sin of Golden Calf

 Or HaChaim (Shemos 30:13) The alienation between Israel and its G'd resulted from the episode of the golden calf which affected the roots of all the Jewish souls. G'd therefore commanded that the people give a "half-shekel" to symbolise that their previous action had resulted in something whole having become divided. The contribution of this half-shekel was to repair the damage done to the whole through the people's participation in the sin of the golden calf. Every mitzva is meant to close a gap that may exist between man and G'd.