This claim probably derives from back-of-the-envelope calculations for cellular turnover rather than "organic material," and is quite inaccurate since some cells turn over very quickly and others last a very long time, making an average a poor summary statistic. I think this explanation for the root of the claim also makes sense given the exclusion of the brain, but likely should be narrowed further to neurons, since the brain is made up of many non-neuronal cells as well. Blackburn doesn't cite anything in making the claim, so to me from a scientific standpoint it is useless; as a philosophical argument it's as useful as the readers deem to trust it.
Neurons in the brain do not turn over; the extent of adult neurogenesis is a bit controversial in humans, but even the strongest evidence makes clear that adult neurogenesis is limited (this Q&A at Biology.SE is relevant).
However, proteins, DNA/RNA, glucose, etc, all turn over at some rate. Like individual cells, the rate of turnover for an individual molecule is incredibly variable. Whole DNA molecules do not turn over. Bases might be replaced due to DNA repair following damage, but most of the molecule remains intact for a lifetime. Proteins themselves have variable life span: some are degraded in hours, others last much longer. The brain certainly is not immune to these processes, but the basic structure is preserved.
Spalding, et al 2005 show that carbon-14 levels in the DNA of neurons in the human brain match atmospheric carbon levels from near their birth (this study is possible due to nuclear weapons testing); effectively, they show that the vast majority of brain DNA carbon does not turn over in the lifetime. By contrast, the 60-70% of the total brain carbon can be considered food-derived after just 58 days in rodents (Buchanan, 1961). Likely this is because much of the brain carbon is not in very long lasting molecules like DNA but rather in short lasting ones that are metabolized and excreted (for example as CO2).
In summary, it's not really a statement that can be analyzed scientifically without being made more specific.
Buchanan, D. L. (1961). Total carbon turnover measured by feeding a uniformly labeled diet. Archives of Biochemistry and Biophysics, 94(3), 500-511.
Spalding, K. L., Bhardwaj, R. D., Buchholz, B. A., Druid, H., & Frisén, J. (2005). Retrospective birth dating of cells in humans. Cell, 122(1), 133-143.
