Rec.humor.funny as the world's oldest blog
Rec.humor.funny was created on Aug 7, 1987 after having first been proposed in 1986. This might qualify it as the oldest still-running blog.
(You can read a bit more of the story of RHF's creation if you like.)
It wasn't the first blog, though. I assign that title to one of the earliest moderated newgroups "mod.ber" which was created by Brian Redman in late 1983. Mod.ber was a newsgroup where Brian and some friends identified interesting threads going on around the net, and posted summaries of them so people could find them -- very much like many of the popular "cool thing" link blogs today.
The debate about the first blog and the oldest blog will of course depend on how you define a blog. Here's the parameters as I am conveniently defining them:
The last point may make you ask why I am talking about usenet newsgroups, which predate the term "world wide web." That's because when Tim Berners-Lee coined the term and defined the HTML and URL standards in the early 1990s, he very overtly declared the web to include the things you could link to from HTML, including a number of the earlier data communications systems. Most notably it included gopher (the WWW's most direct predecessor), telnet login sessions, sending (though not reading) E-mail, finger for live status, FTP for file transfer and USENET newsgroups for serial publishing and discussions.
There are people who don't include these other systems when they think of "the web" but they'll have to argue with the man who defined the term and the protocols if they want to insist on it.
USENET of course was mostly open discussion areas, known as newsgroups there, and "message boards" in most other areas today. However, it has a facility known as moderated newsgroups. Moderated groups are under the control of an individual or small group. Sometimes they are just used for more tightly controlled online discussion, but in a few cases, they were used for publication of journals with a single editorial voice. Mod.ber was the first of these.
Before USENET there were mailing lists, particularly on the ARPANET. The other main online serial publications to predate mailing lists were online community systems like Plato done on single machines, and the "news" systems found on a number of timesharing systems. (USENET, often known as Netnews, was in fact in part created to replace these systems, which kept timesharing users up on the latest developments at their site.)
There are a few other online publications which might also make a claim for being first or oldest blog.
Risks Digest is a superb online publication which began in 1985 and continues to this day. It is more of a forum than a journal, and in fact calls itself a forum, but unlike most online forums it has a mildly applied editorial voice from Peter Neumann.
There is also a popular mailing list known as the Telecom Digest which is still in operation. It dates to 1981 and is perhaps the oldest mailing list still in operation. (Though see about IP) It is also mostly an online forum, though the editor makes frequent comments to add a mild editorial voice.
Dave Farber has the distinction of starting the world's first network mailing list in MSGGROUP, which began in June of 1975. The original MSGGROUP was just a mailing list however, without an editorial voice. It has been supplanted however by Dave's Interesting People mailing list which while still a forum, has a much stronger editorial voice. Dave posts only a small fraction of the messages sent into him. There was, however, a break between MSGGROUP which ended in the 80s and Interesting People which began some years later, to the best of my knowledge.
Of course, if you still insist that only an HTML/HTTP page can be on "the web" most people cite the "What's New" page maintained by the UIUC NCSA, the creators of the Mosaic web browser. This page listed interesting new sites as they came on the web. It may be viewed more as a news page than a personal journal, but many other pages of that sort appeared shortly thereafter. There are some who believe the first "blog" is one that came after the term was coined (late 1997,) but I can't put much credit in that -- the phenomenon predates the term by quite a bit.
Some people would list their daily thoughts and activities in their .plan file which would show up if you used the "finger" program to query them. The finger protocol was proposed as an official web protocol in 1994, and even showed up in some browsers, but never took off. Generally these files were just a single entry. Finger dates back to about 1977, it was created by Les Earnest. I don't believe anybody still keeps up a journal in a .plan file.
(Finger was notable however for being the first program to let you see the "presence" of another person over the net. It showed you if they were logged on and their keyboard idle time.)
There is a report that at CMU a tool was in use in 1980 that gathered .plan files via finger and sent them via a mailing list to subscribers. This could also qualify as the first blog, though E-mail's inclusion in the web is somewhat more undefined. You could always send E-mail via the web (with the mailto: URI) but there was no URI for reading it, as there was for USENET and finger and the like.