Best Technology Available
By late 2002, SSDs had just reached ~80 GB capacities. Of course, JWST is not going to take a product which has never seen enterprise deployment, and plonk it into \$2 billion platform with fingers crossed, hoping the vendor did a really good job. The bleeding edge referred above is for terrestrial, consumer-grade drives. Once we factor in radiation hardening for space usage, existing product history for confidence, and whatever additional redundancy mission designers chose to hedge their $2+ billion toy for its 5-10 year lifetime, 68 GB starts to look pretty reasonable.
Yes, it took 10 years to go from MB to GB, and another 10 to go from GB to TB. So by 2010, we already had TB standalone drives (not arrays) in a COTS 3.5" SATA form factor. But by 2007, the JWST had already passed most of the core reviews, so technology had to have been chosen years prior, using whatever was available then. By 2006, NASA had already shelled out $1 billion on the project. The prime contract was awarded in 2003. It is not hard to imagine that the data storage technology was selected within a year or two of this point in time, given that they had already burned through a billion dollars in development just 3 years later.
If we were to redesign JWST right now, we could choose something like Mercury's 440 GB space-qualified RH3440. This is bleeding-edge technology, and it's not even 1 TB. Also note that it uses more "primitive" Single-Level Cell NAND technology, rather than high-density Quad-Level Cells. Obviously, this is for robustness and part of what makes it space-qualified. This is why you cannot compare consumer-grade and space-grade products on a level basis. Obviously, we could put a few of these in the JWST, and get over 1 TB of storage, but I would imagine that putting a more than dozen of them on board would push size and power constraints. So let's say we could get up to 6 TB of storage for JWST 2.0, 2021 edition.
The exponential growth of SSD density has followed an approximate ratio of 1000x per 10 years. Rewind 10 years to 2011, and we would expect to plop a mere 6 GB on board using the cutting-edge space tech available at the time. Go back 17 years to 2003-4, and the fact that it has 60+ GB of SSD storage is actually looking pretty remarkable, given that we would extrapolate it to maybe 60 MB (space-grade!!!). There may be well more than a dozen discrete drives on board JWST (can't find design details at that level of granularity).
The real question is not: "Why is there so little storage?" but rather: "How did they get so much on board?" Perhaps they were allowed to cheat and update the storage later in the design process to newer but still mature technology. The Critical Design Review, which I imagine cemented a lot of decisions into stone, occurred in 2010. If they got to harvest 6-7 years of advances, that could explain a 100x improvement over expected capacities.