The term, according to Wikipedia, was coined on September 10, 1999 by Brad L. Graham, as a joke.  (http://www.bradlands.com/weblog/1999-09.shtml) It was re-coined in 2001 by William Quick (http://www.dailypundit.com/) (quite seriously) and was quickly adopted and promulgated by the warblog community.
In fact, Brad’s entry is worth reproducing here at length:
IT’S PETER’S FAULT: A year ago, “weblog” was hardly a common word… Then the supremely urbane Peter Merholz decided it would be fun to pronounce “weblog” as “wee’blog” and I thought that was kind of cute. Then folks started truncating that to merely “blog” and — ugh! — it’s stuck! …
So, now then. Where are we headed? Will personal publishing soon be described as being “as simple as falling off a blog”? Shall we see ultra-conservative gays start weblogs and dub themselves Blog Cabin Republicans? Track the tides with an Ebb Blog? Is blog- (or -blog) poised to become the prefix/suffix of the next century? Will we soon suffer from (and tire of) blogorreah? Despite its whimsical provenance, it’s an awkward, homely little word.
Goodbye, cyberspace! Hello, blogiverse! Blogosphere? Blogmos? (Carl Sagan: “Imagine billions and billions and billions of blogs.”)
So that’s where it all started. What I find interesting is that although the term reappeared, and was ‘formally’ adopted, in 2001, mainstream coverage didn’t begin until 2002, according to my tireless research on Factiva. Indeed, it didn’t really take off as a mainstream term until late last year, when the U.S. election put the whole blogging thing into the political spotlight. See how mentions remained relatively flat (at between 20 and 50 a month) before suddenly taking off in October 2004. Mentions haven’t really dropped off since:
Sadly now it seems we’re stuck with the term. Suggestions for a better word?