File systems and databases are useful abstractions, from an ease of use, security, and robustness perspective. The challenge is to make them go faster. Pushing the machinery behind them out to special-purpose hardware can do that.
The straightforward thing to do first is to to take some FPGA part and use it to implement a large key/value store using non-volatile solid state memory. That's been done at Stanford, Berkeley, and MIT, and was suggested on YC about six years ago. One could go further, and implement more of an SQL database back end. It's an interesting data structure problem; the optimal data structures are different when you don't have to wait for disk rotation, but do need persistence and reliability.
Going back to non-volatile memory the question is what kind of abstraction should be implemented in hardware? Presumably something simple that the OS and applications can then use to implement higher level abstractions like file systems and databases. Pushing parts of a SQL database engine into the hardware does not intuitively seem like a right solution.