The term 'old mine' refers to gemstones deriving from historical, often ancient, mining sources.
With regards to diamonds, for example, this would indicate that the stones were mined in India or Brazil, the two primarily mining locales prior to the discovery of diamonds in South Africa in the late nineteenth century. India in particular is renowned for diamonds mined in the central area of the subcontinent surrounding the city of Golconda, and are known for their particularly high quality in terms of size, colour and clarity. The Hope diamond and the Koh-i-Noor, perhaps the two most famous diamonds of all time, are Golconda diamonds. To add to the rarity of such stones, these mines have been largely exhausted.
In the case of emeralds, stones classified as old mine would be from Egypt, Austria, Gandahara (the ancient central Asian kingdom) or, primarily, Colombia. The highest quality old mine emerald material derives from this last source, which had been mined by pre-Colombian tribes for approximately one thousand years. It was not until the Spanish presence in the New World in the early sixteenth century, however, that these stones graced the royal courts of Europe, the Middle East, and, primarily, India.