Many sources reveal that before "Google Flu Trends", "Found Data" was the term used in many companies {Found Data: the digital exhaust of web searches, credit card payments and mobiles pinging the nearest phone mast}.
"Big data" and "found data" might have much in common, but I wonder if a distinction may be that found data could be something someone sought after, whereas big data might too often be a pile of data we collect, with no clear purpose, and then just see what associations we might run across. ??? (I am not an expert on this. Does a 'found' data set have to be large?) In that case, found data may be superior in that there may have been a legitimate scientific reason/model behind the search for relevant data, whereas when big data is about nothing but associations, we might incorrectly infer causality.
There was a lively thread under multiple linear regression not long ago, where it was made clear (notably by Theo Dijkstra, but it was a consensus) that when determining what regressors to keep, there should be a reason for modeling that way, not just relying on the data. With big data there may be less question about one batch of data leading in a different direction than another, as that could be tested, but we could still see spurious relationships. [A dermatologist once told me it was unusual for someone with such dark hair to have skin cancer. What he missed was that I have dark hair, but pale skin. You can't just look for associations in a vacuum without sometimes going wrong. Perhaps that is not exactly the right analogy here, but I like that one. :-) ]
If, however, found data were just whatever someone 'found' that supports their view ... ignoring data to the contrary ... then big data sounds better.
I imagine, however, that 'big data' and 'found data' can each mean different things to different people under different circumstances - so sometimes they may mean the same thing. (Similarly, it was obvious on another RG question that "representative sampling" meant different things to different people.)
But I think that whatever the name or variation on a theme of a large data set, one must still back up 'found' associations with theory, and much better to theorize first, rather than try to justify after-the-fact, though theorizing first, and then only using data that agree would certainly be as bad.