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A modified coin would be valueless. Any value would be associated with the necklace.
YeS
You could wrap the coin in gold or silver wire and form a loop to pass the cord through. You could find a mount, called a finding, of the right size; these come in every imaginable shape and size, and are meant for mounting polished stones into. You could embed the coin in acrylic plastic and drill a hole in the plastic. Doubtless there are other methods that wouldn't destroy the coin by drilling it.
Such a coin is fairly common and is not made out of any precious material and since it is in a necklace, there is no coin collector value. Any value would come from the necklace itself, most likely scrap silver/gold if the chain or bezel was made out of that. In short, the coin is worthless and the only value comes from the necklace part itself.
If you have a 20 dollar gold coin, it is worth a lot, at least $1500. You need to know what year the coin is from.
http://www.limabeads.com/Green-Girl-Pewter-Mermaid-Pearl-Coin-Link-32x23-P5704?ref=froogle that kind of looks like the coin except in silver.. and you could probably add the pearl hope i helped!
You will find that the necklace and earrings are attached to a bust of Queen Elizabeth II. All British coins issued from 1985 to 1997 depict Queen Elizabeth II wearing a necklace and earrings.
Take it to a coin dealer
The coin is worth very little, maybe £1 if it was in absolute mint condition. Any value would be associated with the necklace.
SIMPLE ANSWER: No US coin dated 1965, 1966 or 1967 will have a mintmark, it's just a penny.
A coin flip
It's not possible to tell without knowing the coin's denomination and whether the holder is actually connected to the coin. If it was soldered on, for example, it would damage the coin and reduce its value somewhat. The good news is that gold is now so high in price that the coin is probably worth more for its metal content than as a collectible.