Herdotus claims the first coins were minted by the Lydians. Aristotle writes the first coins were minted by Demodike of Kyrme, who was the widow of King Midas of Phrygia. Modern numismatists now think the first coins were minted on the Greek island of Aegina. The first coins from Aegina were issued by the local oligarchy, but soon the mint at Aegina was producing coins with the image of King Pheidon of Argos on them. The problem is, this period is poorly documented. About the only firm date we date we have for King Pheidon of Argos is when he defeated the Spartans in 688 BCE. There are no firm dates for the King Midas of Phrygia that was married to Demodike of Kyrme, although he was probably a near contemporary to King Pheidon of Argos. So modern numismatists can not say for certain who was the first ruler that had their face stamped on a coin, but right now it could be either King Pheidon of Argos or King Midas of Phrygia. Their coins have no dates, but they were minted sometime around 688 BC.
Source: http://www.ancient.eu/coinage/