The origin of the use of the "Half-Shekel" coin is to be found in the Bible, Exodus 30:11 -1 5.
"The Lord spoke unto Moses: When thou takest the sum of the children of Israel This they shall give... one half shekel The rich shall not give more and the poor shall not give less than half a shekel when they give an offering." The tax was equal for all, rich and poor, because every person is equal before G-d. Half a Shekel and not a whole Shekel signified that a person cannot obtain completeness alone.
Today, on the Shabbat before Rosh Chodesh Adar, known as "Shabbat Shekalim", the special "Shekalim" portion of the Torah is read. On the eve of Purim, the equivalent of a Half Shekel is donated in commemoration of the original Half Shekel Tax.
In the year 5721-1961, the Bank of Israel issued a special Half-Shekel coin (Half of an Israeli Lira) which was legal tender in the State of Israel. Bearing a face value of Half an Israeli Lira, it became an appropriate coin with which to perform the commandment of commemoration of the "Half-Shekel".