The really interesting question, to me, is why is it so important? In addition to spawning threads on JREF, Dawkins spends several pages in The God Delusion on the subject. People seem to have strong opinions.
In the sense used in every math textbook in the world, evolution is very clearly random. In almost every sense of the word found in the dictionary, evolution is clearly random. There is one and only one sense found in the dictionary in which evolution is not random. That's the sense of "all outcomes being equal". That clearly is not the case with evolution, but neither evolutionists nor creationists say that it is. We can disregard that sense because the context makes it clear we are not using that sense.
Usually, JREF is pretty sympathetic to using the technical, mathematically precise, definition of a word. Why is this one different?
As usual, I'll offer my opinion. It is partly because the term has occasionally been misused by creationists. Some of them say "evolution claims that animals are assembled at random." Evolution doesn't say that, and some scientists (both professional and amateur, like us) bristle at the comparison. The assembly of organisms follows instructions in the DNA, and the DNA itself, while the result of a random process, is assembled from its predecessor version of DNA, with perhaps a few random errors. No molecule of DNA, except possibly the first, very simple, molecule, has every been assembled randomly.
However, I don't think that's the full explanation. Despite the misuse of the term, the actual term, when used correctly, is very precise, and very accurate in describing evolution. To give my opinion, I'll turn to Dawkins' explanation from "The God Delusion".
Actually, it's from "Climbing Mount Improbable", but I haven't read that book. I've only read the shortened version of the argument in "The God Delusion". In that book, he describes evolution with an analogy of someone staring at a cliff, and saying, "No one could ever go up that cliff!" The "cliff" is the assembly of complex organisms from simple molecules. However, on the other side of the cliff is a gentle slope, that can be walked up easily. The gentle slope is the slow variation of organisms over time. Each step is quite easy, and with time, the "summit", a complex organism, is reached through that series of small steps.
Every analogy has weaknesses, and we can see what Dawkins is getting at here, so the analogy works. However, I think it is not a very good one. I think a better one would be to replace the gentle slope with a pockmarked, canyon strewn wilderness. It is filled with craters, small cliffs, landslides, low summits and deep crevases. Organisms grope blindly in all directions all over the wilderness. Eqrthquakes reshape that wilderness periodically. In the wilderness, though, there is a path leading to the summit. It goes up and down all over the place, but it does get there, and it gets there with no single step that requires a large cliff. Lots of organisms start at the bottom, and their descendants end up in every nook and cranny of the wilderness, including, eventually, one that finds its way to the top, after many, many failures along the way.
I think that's a better analogy, but it isn't very appealing. It makes it seem like the achievement of the summit isn't an "achievement" at all. Eventually, by sheer dumb luck, someone will make it to the top. That isn't as exciting as saying that the complex organisms, i.e. us and our pets, are part of a magnificent adventure that makes clear the amazing power of nature. Nevertheless, it is accurate. We have achieved a temporary summit on the evolutionary hill, and we did it by chance.