Why is a 1965 silver dime worth a lot of money?

There are unconfirmed reports of a few 1965 dimes that were struck on leftover silver blanks, similar to the way some 1943 cents were struck on bronze blanks.. IF one exists, it would be valuable because it would have been struck in a metal that had not been used in dimes for well over a year.

When the price of silver started to rise in the early 1960s coins started to become worth more for their metal content than their face value. People were hoarding silver coins and selling them to metal dealers who would melt them. You could buy a $10 roll of quarters at a bank and sell it for $20 (or even more) to a metal dealer, for example! As a result the government was taking a loss on every coin minted and there were serious shortages of coins for making change.

To stop the losses the decision was made in early 1965 to make coins out of the copper-nickel "sandwich" material that's now used in all dimes, quarters, and halves. For a while in 1965 the Mints struck 1964-dated coins out of silver while the new metals were being prepared, so there is the possibility some silver blanks got mixed in with copper-nickel ones.