Came across this when one of my deliveries was late, can be used up to 12 times in a one year period. Contact customer support (I opted for by email), let them know that the delivery is late, and ask for a free month. Thanks to Dealnews.com for the info and the wording to use in the request. I have used this twice this month for 2 month extension to my Prime subscription.

"Hello! My order, #XXX-XXXXXXX-XXXXXXX, was scheduled for delivery on the 18th, however, it has not arrived. I saw that Amazon will extend a Prime membership by a month if a package misses its delivery date. Am I eligible for that extension for this order not arriving on time?"

Details here:

https://www.amazon.com/gp/help/customer/display.html?nodeId=201117450 (note at bottom of page)

And here:

http://dealnews.com/features/How-to-Get-1-Month-of-Amazon-Prime-for-Free/938946.html