The most that can be said is that it wasn't mere leprechaun gold.
The story of a leprechaun with a pot of gold needs no explanation, and this appears to be the basis for the infinite Hoard of coins that Sweeney can pull from anywhere, but the story of a leprechaun with a single special coin... isn't really found easily.
Actually, Sweeney says it himself:
I did it all like he said, but I gave you the wrong coin. It wasn’t meant to be that coin. That’s for royalty. You see? I shouldn’t even have been able to take it. That’s the coin you’d give to the king of America himself. Not some pissant bastard like you or me.
Sweeney himself is implying the coin isn't even supposed to be something he can access. That carries the implication that it wasn't part of his legend, his myth, his story - we know that's a bit possible, as this is a book mashing all legends together. Whoever's coin it was supposed to be, Sweeney now has to pay a price for, and can't.
The gold coin is never actually described in detail at all - we're not told what's on its faces or etched on the edge. Shadow mentions it's about the size of a half-dollar, that's all we get. Since the other similar coin in the story is mentioned to be a 1922 U.S. silver dollar, I took a brief run through coin history. It tells me that no American gold coin was ever minted in that size - most were made in the 1800s and were smaller than dimes (for non-Americans, the dime is the smallest current coinage, and the half-dollar the biggest). Sweeney describes it as a golden sun, but never makes clear if that's actually what's on the coin.
The only other input we get on it seems metaphorical at best, but is from Zorya Polunochnaya:
You were given protection once. You were given the sun itself. But you lost it already. You gave it away. All I can give you is much weaker protection. The daughter, not the father. But all helps. Yes?
What legend she is referring to is not really clear, but the father of the Zorya sisters in Slavic myth is Dažbog, one of several possible sun gods in that pantheon. This really doesn't connect that deity to the gold coin however - Zorya Polunochnaya herself doesn't represent the moon that she gives to Shadow, but a "midnight star", and is regardless a character that Gaiman admits to inventing completely (there's some debate, but there's usually only two Zoryas in this mythology). So her metaphor can't really be applied literally to the characters. Credit to @Adamant for pointing this link out in comment.
A list of possible sun deities the coin could link to is easily found but not easily analyzed - a great many are tied to life, or rebirth, or other life concepts, in some way. The coins could each represent more general worship of the sun and moon... but this isn't normally how objects of worship are represented in this setting, they're usually more human-looking than coins, so that's a stretch at best.
An alternate interpretation is that the coins aren't deity-myth related, but American-myth related. The phrase "life, liberty, and the pursuit of happiness" is lifted from the Declaration of Independence, and often used as catchphrase for America's identity. This is borne out by the silver coin - it's literally stamped with Lady Liberty, it's presence is brought up prominently every time Shadow is held captive, and Zorya Polunochnaya tells him in the afterlife that "It bought you your liberty twice." The gold coin would then represent life - both the old life that Shadow gave up on tossing away the coin, and the almost-life that Laura receives from it.
This is unfortunately an endless fountain of speculation, and there does not appear to be any solid answers.
In the Starz series, the coin is very obviously old and stamped with the sun. So while not an American coin, the subtext of life and rebirth from the book are still very obvious.
The show also builds on his legend. In his highlight episode A Prayer for Mad Sweeney, Sweeney holds a conversation with an imprisoned Essie. He tells part of his own story involving delivering a tithe, a share of his gold, to a king. Which king is never elaborated upon. The obvious implication is that the one special coin out of his endless horde represents the tithe that is owed.
Since in the show Sweeney deserted a war, and presumably any loyalties he had, it's possible that the sun coin was the only shred of his duty and myth left, that fortune continued to work for him because he was still intending to deliver it. With the coin lost to him, and Sweeney's luck turned (bloody hilariously) bad, fortune is collecting it's due with interest.
It's also subtle foreshadowing. Put this addition to his story together with his outburst from the book (the series hasn't yet reached this point, so it may change) that he "shouldn't have even been able to take it", especially with how nonchalantly he was tossing gold coins everywhere to screw with Shadow.
It points to the fact that Shadow, as the literal son of a god, qualifies as royalty, which is why Sweeney was able to lose the coin. Although as he's not a king of anything, he certainly wasn't the royalty intended.