While coinage likely emerged in several places independently, ancient Greek writers did claim to know who the inventor of coins was. According to Aristotle, the wife of Midas (of golden touch fame) of Phrygia (a kingdom in what is now geographically Turkey) named Hermodike of Kyme was the first to strike coins. This would have been about the 8th century BC.

This is semi-mythological, but the first archaeological evidence of coins does come the same region, in the Kingdom of Lydia which rose after Phrygia and was eventually conquered by the Persians.

Source: Muscarella, Archaeology, Artifacts and Antiquities of the Ancient Near East

According to Peter Bernstein's The Power of Gold: The History of an Obsession, the first person credited with establishing currency was King Croesus of Lydia. His capitol sat on the river Pactolus, which was rich in natural gold and silver deposits. The coins (stamped with his face) were the first to be recorded as "standardized," with an official mint and relative standards of purity. Later coins were more refined gold, while earlier ones were made of natural deposits of electrum, a gold/silver alloy. If you've ever heard the phrase "rich as Croesus," this is where it comes from.

Unfortunately for Croesus, his line didn't survive to found any kind of family, banking or otherwise. Instead, his Lydian kingdom was taken over by Cyrus the Great of Persia in 546 BC and he and his family were put to death (although there are legendary stories of his miraculous survival.)