He didn't invent these “6 wierd things about me” blog posts as such, rather the meme was introduced in his first best-seller The Selfish Gene as an example of the power of self-replication.
The book is, of course, about evolution and describes the mechanism of genetics: how processes that promote the propogation of almost perfect copies of genes from generation to generation are more likely to continue than other processes. It is a well-written, entertaining, rigorous read that attempts to share some of Darwinism's beauty and magic.
At one stage in the book Dawkins wanted to make the point that genes need not be the only type of replicator that does that and suggested that cultural ideas act in pretty much the same way. Popular songs, views on Julia Gillard, jokes about Damir Dokic: they live in one person's mind and can be passed on to the mind of the person next to him through the mechanism of speech.
As with genes, some memes are better at propagation than others. Dawkins elegantly describes the example of the lyrics to Auld Lang Syne. The refrain is almost universally sung ‘For the sake of auld lang syne’ when Burns actually wrote ‘For auld lang syne’. Why has this mutation survived so well in the meme pool? Dawkins suggests that as most people learn this song by listening to other people singing it rather than by looking it up in their copy of the Scottish Student's Songbook. If you are unsure of the words and are nervously singing along with a large group of people belting out this old classic and, no matter how loudly the people who actually know the correct words are singing, if there is even only one person erroneously throwing in the ‘for sake of ’ bit, you will hear it loud and clear — the S and the K really cut through the sound and that's what you hear and, thus, you think that's how the song goes.
This idea of a cultural unit acting as replicating entity entered the language shortly after the publication of Dawkins's book (acting memetically as he himself later noted) and some blogger must have applied the word to this idea of taking some rules about what to post and passing them to one of your blogfriends. The rules are replicated, you understand.
Anyway, for INC's pleasure I'll quote the paragraph where it makes its first appearance:
We need a noun for the new replicator, a noun that conveys the idea of a unit of cultural transmission, or a unit of imitation. ‘Mimeme’ comes from a suitable Greek root, but I want a monosyllable that sounds a bit like ‘gene’. I hope my classicist friends will forgive me if I abbreviate mimeme to meme. If it is any consolation, it could be related to ‘memory’ or to the French word même. It should be pronounced to rhyme with ‘cream’
And to join in with this particular meme I'll say that one of the wierd things about me is that I tend to write pompously when the subject turns to something I once read about.