Definition: Barrel aged cocktails are one of the latest trends in mixology in which pre-mixed drinks are placed in barrels for a few weeks. The technique's purpose is to change the cocktail's character and flavor, mellowing the mix in the same way wines and distilled spirits are aged.
The mixologists behind aged cocktails are Tony Conigliaro of 69 Colebrooke Row in London, who has been aging in glass, and Jeffrey Morgenthaler, who brought the idea to the United States and began using barrels. Morgenthaler's experiments with various barrels, aging times, and recipes are best read in his own words, which can be done on his blog: Barrel Aged Cocktails.
As Darcy O'Neil points out at Art of the Drink (blog post: Barrel Aged Cocktails), aging drinks - or at least storing pre-mixed drinks - is not new. If you have checked out Jerry Thomas' Bartenders Guide you may have noticed that the 19th century bartender made record of a "Bottled Cocktail" - essentially a large batch of Brandy Cocktails and includes recipes for a gin and a bourbon version - and a number of essence and punch recipes specifically for bottling. I agree that this technique was likely more out of convenience than for the purpose of allowing the drink mix to take on different taste. Punches in particular are often best made in large quantities and in that day and age they were very popular. By having prepared mixes in stock, Thomas could have a punch ready for any patron who wanted one.
If you are interested in learning more about aging mixed drinks or want to try it yourself, I recommend reading through both articles referenced above.