It's been a little while, and the Crazy Australian has in fact conceded that chance can at least produce something which we recognise as information. This Australian says, good on him.
This question is a common creationist approach. The application there is that information in a genome is not seen to increase after random recombinations. The response is that new information can take the form of a decrease in data as well as an increase (my usual example is that the valuable information each week on Big Brother is who will leave the house) so a recombination, which is partly both, is definitely an increase in information.
Completely random threads of information would indeed be unlikely to come to anything meaningful. Consider that, by the way: information, as defined in a computational sense by those who wish to manipulate, compress and optimise it, need not be meaningful in any sense other than representing the data of which it is comprised.
A bit of real-world pressure makes all the difference; this is the very core of natural selection. Changes to a genome which are harmful, ambiguous or negligible tend to die out when the living beings carrying them are outbred by beings carrying directly beneficial changes. It doesn't matter what the information in the genome means as such. Only survival value is of importance, and as it happens, information coherent enough to represent blueprints for complex structures like wings and stomach acid has substantial survival value.
You know, the real answer to this question is really extremely simple. Yes, chance can produce information, if there is such a thing as information. Chance can produce anything, given the right conditions. That's what chance means.
Applied to anything real, though, the true question is, "Can chance consistently produce information?" That's a little hairier, but it's moot when applied to evolution. Natural selection is not a chance process. It mines the random element of mutation for new genetic material, but the selection itself is the original Survival of the Fittest.
It's like rolling fifty dice and only picking out the fours, fives and sixes. You're guaranteed a fairly high average score per selected die. The game of evolution is rigged in favour of life.