Remember "... it's the Cover-up"? Well what if the reason for the tweet was

exactly that! How better to cover-up your fishing expedition for the "hidden and widely distributed evidence" that Obama scattered throughout various National Security departments than to tweet a made up claim so Trump could then use as cover to start searching for all the copies of the supposed evidence?

Even more ironic would be if Obama hadn't chummed the waters at all. If it was simply a rumor placed to start his frantic searching to cover it up and thereby expose exactly what investigators needed to look for. Even if it doesn't lead to anything it shows the behavior of the guilty party that can be used to corroborate evidence collected by other means.

Oldest trick in the DEA's book.