What makes a coin more valuable than its original circulating value?

Answer 1

Various factors can make a coin worth more than its face value.

Older coins that were made of gold or silver are worth more simply because those metals cost much more per ounce than they did when the coins were first issued. For example up till the early 1960s the price of silver was controlled by the government at roughly $1.29/oz, so a silver dollar with 3/4 oz of that metal in it was guaranteed to be worth $1. Once silver was deregulated its price increased significantly, so those coins that once held only a buck of silver could be profitably melted at 10 or 12 times as much.

Other coins may be worth more due to scarcity and/or collector interest. As an example, there was also a major public controversy over the presence of the designer's monogram on the first Lincoln cents minted in 1909 that forced the Mint to redesign the coin part-way through the year. By that point San Francisco had struck fewer than a half-million cents with the monogram, so the combination of notoriety and relative scarcity drove prices far above 1 cent in just a short period.