The Oxford English Dictionary defines blog as
A frequently updated website, typically run by a single person and consisting of personal observations arranged in chronological order, excerpts from other sources, hyperlinks to other sites, etc.; an online journal or diary;
—and then notes that the term blog is in more common use than its etymon, or parent word, weblog.
The earliest quotation the OED gives in association with the etymon weblog is from 1993:
comp.infosystems.www (Usenet newsgroup) 10 Nov. (title of posting) Announcing getsites 1.5, a Web log analyzer.
This example though does not really point to the source of blog; rather, it’s an example of the OED’s first definition of weblog:
Usually as two words. A file containing a detailed record of each request received by a web server, frequently recording data that allows a variety of different aspects of the web traffic reaching that server to be analysed.
This definition differs from the second definition the OED gives for weblog, which is synonymous with the definition for blog above.
A quotation given by the OED under the entry for blog from 1999 points (somewhat humorously) to the sundering of web and blog into a new word:
[1999 http://www.bradlands.com (blog) 23 May (O.E.D. Archive) Cam points out lemonyellow.com and PeterMe decides the proper way to say ‘weblog’ is ‘wee’- blog’ (Tee-hee!).]
The OED attributes the PeterMe blog (by Peter Merholz) with a more direct citation for the word blog, dating from 1999:
For those keeping score on blog commentary from outside the blog community.
(A 1999 entry cited from the Edinburgh Scotsman cites the word with an apostrophe: ‘blog).
Blog as a verb, as well as blogger and blogging all get citations going back to 1999.
The Online Etymology Dictionary gives the following entry for blog:
1998, short for weblog (which is attested from 1994, though not in the sense “online journal”), from (World Wide)Web (n.) + log (n.2). Joe Bloggs (c. 1969) was British slang for “any hypothetical person” (compare U.S. equivalent Joe Blow); earlier blog meant “a servant boy” in one of the college houses (c. 1860, see Partridge, who describes this use as a “perversion of bloke“), and, as a verb, “to defeat” in schoolboy slang. The Blogger online publishing service was launched in 1999.
Weblog is of course a portmanteau of web and log, both of which are abstract and concrete, new and very old. In The Origins of English Words: A Discursive Dictionary of Indo-European Roots, Joseph T. Shipley gives the root of web as uebh:
uebh(s): weave, move back and forth; objects woven or the like, as a honeycomb. Pers, baft: woven cotton cloth. Gk huphe: web. hypha: threadlike part of fungus
Shipley gives over a dozen other examples that generate from uebh including hymn, hymen, vespa, wasp, weaver, woof, waffle, wave, and gopher.
Shipley’s root for log is presumably leg I (he doesn’t list log in his Index of English Words). Shipley gives his definition of leg I:
leg I: gather, set in order, consider, choose; then read, speak. Gk, logos, logion, horologe, horology, lexicon.
Shipley gives leg I as the source of many words, but helpful to our etymology of blog are intellect, illegitimate, select, legend, sacrilege, sortilege, collect, and cull.
The entry for leg II is of course beneath the entry for leg I. Shipley points to this as the root of (among others) lax, laxative, delay, relay, languid, languish, lash, lush, profuse, leach, leak and lack. Any of these will fit into a proper etymology of blog too, I suppose.
Biblioklept 
Hah.
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