I already know historians who have encountered these problems, specifically dealing in biographical studies of artists. (I mean “artists” in the broad sense – writers, musicians, etc. as well as visual artists.) When they go looking for documents from the 1950s or 1960s, they find oodles of stuff, often saved both in personal collections of papers and in various archives.
But I know someone who was recently working on an artist who died only a few years ago, with a career spanning much of the late-20th century. Suddenly, there were huge gaps in correspondence popping up starting in the mid-1990s, because everything was handled by email, and it hadn’t been archived anywhere. (At least not anywhere in his personal collection; I suppose some future historian might find some of it in some server archive, if they ever get access to such personal documents.)
And, in reply to those who just assume stuff will get archived, some early internet stuff has already disappeared – heck, even stuff from recent years. Look what happened when GeoCities shut down in 2009. Suddenly, 15 years of user contributions to the internet was taken offline. A few projects engaged in last-ditch efforts to try to archive as much as they could, and apparently the Internet Archive got a lot of stuff, but that’s just one (big) example. There are so many smaller services and sites that have disappeared over the years, probably without a trace.
On the other hand, of course, there will still be loads of crap to sift through. The issue, from my perspective, is more about personal electronic documents, rather than public ones. Even if the data is preserved on some server somewhere, will future historians be granted access to data that was one password-protected? My guess is that, at least for the next couple decades, a lot of that data will just disappear from servers and archives before legal hurdles and other issues are dealt with to preserve and ultimately allow access to such stuff.
Another interesting issue that historians already deal with in the digital age is the loss of drafts. People who study writing or music composition or other such artistic endeavors often learn a great deal from looking at drafts, which in the past were often handwritten or at least typewritten. These days, a lot of writers and composers just correct files on the screen as they go, and we have no record of the creative process.
Of course, in the past, there were always people who burned their drafts or papers in general. And in the distant past, we rarely have any of those records. But for people doing recent biographical history, the 1950s is often much richer in surviving accessible personal documents than the late 1990s.
I think ultimately many of these problems will be dealt with, but future historians will probably note an interesting gap in certain kinds of resources beginning in the 1980s or 1990s and going for a few more decades into the future.