Coin age is a way to display how long a coin has been in one's possession in order to prioritize it for use in transactions or mining. It is calculated by multiplying the number of coins by the average amount of time in blocks they have been possessed. For example, 3 BTC that have been possessed for 4 days would have a coin age of 12 BTC-days (3 x 4 = 12).
Under Proof of Stake, possessing a higher coin age increases the chance of mining a block. Giving priority to coin age over volume of transactions would increase the demand to perform confirmations quickly.
Under such a structure, coin age would become valuable.