The problem with the article, and indeed this discussion, is that the factors influencing biodiversity are extremely complex. As scoff noted, the geology of the region in question has a lot to do with biodiversity, as does the availability of niches (heterogenous vs. homogenous habitat, for instance). Observations have shown that niches seem to be more finely divided in the tropics than elsewhere, but the causal explanation is still wanting.
In the case you mention (the effect of genetic drift on small populations) rate of evolution may increase, but rate doesn't necessarily increase biodiversity - which is the heart of my quibble with the article. Take the case of the Galapagos as a for-instance, which scoff mentioned. It is actually a biodiversity-impoverished area overall. However, it has a very high level of endemism and both speciation and incipient speciation, indicative of a high rate of evolution. The islands are only about ~10 my old. Part of the explanation is geologic, some is topologic and climatic, some is ecological (dispersal ability, colonization sampling error, etc).
Mutation, of course, is the primary means of increasing inheritable variability in a population over time. It therefore stands to reason that whenever you see a high rate of endemism and/or speciation, the rate of mutation is also high. The cause of the mutation rate can be argued, as can the relative contribution of drift.
I think another issue we may be having trouble with is the use of the term "biodiversity". How you measure this is problematic, with a number of differing definitions leading to variant results. For example, in my work we usually use "total number of plant species" as the yardstick. This is consistent with the IUCN's definition, and is how global biodiversity hotspots are determined. Another widely used (and useful) definition is the rate of endemism. As you can plainly see, if we use the endemism scale the Galapagos show very high biodiversity, whereas if we use the species number definition, Galapagos is impoverished. I suppose as long as everybody knows which measure is being used, there isn't a problem.