Data exhaust is the trail of information generated as a result of people's online activity. It's exhaust. Fumes. Bad stuff. Stuff that can kill. Or at least indict.
Data exhaust comprises more or less a cloud-shaped form uh, digitally...and it remains so amorphous, it really can't be mined for any kind of meaningful analysis. For example, there is primary or “big” data that relates to the products and services of a business. That data comprises things like, "How many pages into the subscription suite did the customer get before we lost them?" That's structured data.
Data exhaust...ain't. It's basically everything else that's created along the way. A grocery store’s primary data might include which customers purchased which products, while secondary data or data exhaust would tell you what time of day they shopped. Vastly harder to make sense of the purchase of eggs bought at midnight versus noon than knowing the fact that Mrs. Thimmelbottom bought 14 pounds of dog food for Mr. Poopenshnauz.