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.
10 cents for the copper and nickel underneath the gold plating. The U.S. has never made dimes out of gold. After all, if you think about it for a second or two, with gold at roughly $900/ounce and a dime being about 1/12 of an ounce, even the most boneheaded government bureaucrat wouldn't authorize a circulating dime that's worth $80!
The term "5 dollar dime" is confusing. Dimes from 1997 are worth 10¢. If you have a normal Roosevelt dime that has been gold plated, it's still only worth 10 cents. If you are referring to a dime-sized coin that has a $5 denomination, please post a new question indicating the country that issued it and a description of any wording on the coin. That would make it much easier to ID.
Numismatically, the 1943 Mercury Dime is fairly common, and sio would need to be in one of the higher uncirculated grades to be worth much over its "melt value" - plated (even with gold), it will never be worth more than the value of its silver. With silver at $30 per troy ounce (as of February 2011), it is worth about US$2.15.
Ten cents. It's an ordinary dime that was plated. The last circulating gold coins were made in 1933. The lowest denomination US gold coin was the minuscule $1 piece minted from 1849 to 1889. These coins were even smaller than dimes and were worth 10 times as much. A modern dime wouldn't have ever been struck in gold.