Originally, why were half dollars minted?

Answer 1

The same reason that all other denomitions were minted: there was demand for it.
Bear in mind that before the Civil War, paper money was issued by individual banks and not the federal government. As such, paper money was not trusted - often for good reason. Silver dollars were not minted in large quantities until 1878. The 'Silver Dollars that did circulate up until 1878 were largely Mexican 8 Reales, which were worth One Dollar at the time.
The 50 cent coin was the largest US silver coin commonly found in circulation from about 1820 until the Civil War.

Answer 2

They were woth somthing at the time . Same reason they minted half pennies and tuppence (2 cents) etc

Now most of the change in your pocket is pretty worthless