Answer:
Half-dimes were part of an original attempt to create a 10-based US currency system where each major denomination would be 10 times the next smaller, and each would have a "half-" unit to fill in the gaps. E.g. there were cents, dimes, dollars, and eagles, along with half-cents, half-dimes, half-dollars, and half-eagles. Cents and half cents were copper, half-dimes to dollars were silver, and higher denominations were gold. That scheme meant that the half-dime served the same purpose as today's nickel (i.e. it was worth 5 cents) but because it was silver it was very tiny and wore out quickly.
As time went on people got into the habit of thinking in dollars and cents only so the original 10-based system fell into disuse. By the end of the Civil War minting technology had improved to the point where nickel, a much harder metal, could be used for coins without breaking coin presses. It was decided to replace the half-dime with a new nickel coin that would be more convenient in size and last longer. The Mint also adopted metric measurements for coin production, so the new 5-cent piece was also the first metric coin, weighing exactly 5 grams.
The Mint kept up half-dime production for a while longer but the new "nickel" proved to be very popular, and soon replaced the half-dime in daily commerce. The last half-dimes were minted in 1873.