How much is an 1877 gold dime worth?

The U.S. never minted dimes out of gold. In any case, just like today the price of gold was a lot higher than the price of silver and a dime-sized coin struck from gold would have been worth much more than $1.00.

The coin you have has either been plated for use in jewelry, or was exposed to some chemical or heat that oxidized its silver. The bad news is that whatever happened has reduced the coin's value as a collectible. Untouched, it might have retailed for somewhere between $10 and $30. But as a damaged coin it would be worth half that or less.