Whilst this would of course be technically possible I see a number of potentially difficult areas that you would need to address:
- Desoldering the BGA in a manner that doesn't destroy it;
- Finding (or designing) a board which can read the IC;
- Resoldering the BGA in a manner that doesn't destroy it;
- Interpreting the data you actually get out of it.
For (1) and (3) you could use hot air to desolder, then either reball and put onto a new circuit or very carefully solder tiny wires to the pads. Note that although it's a 63 pin device over half are not connected. If you don't know how to do any of these then it's likely a non-starter.
For (2) as the chip is a standard part by a decent manufacturer it is conceivable that you would find another device that uses the same chip. However this device isn't likely to let you read it easily and might actually trash the contents which would be unhelpful.
Better would be to interface it to a small processor (e.g. an ARM or PIC) and implement the right code to read out the contents. With the excellent datasheet you linked this would be possible but it is still not trivial. See this question for somebody who has done it with a different NAND device.
And even if you get this far then (4) is likely a showstopper. Without details of how the original PDA used the memory you will be left with a bunch of binary data that is almost impossible to interpret.
At the highest level you'll be battling however the PDA divided up the internal pages. Below this will be some form of filesystem which has metadata (information on the names of files, their sizes, locations, flags) and contents (the actual file data). The PDA will probably also use some of this NAND for an operating system and configuration, perhaps in the filesystem but perhaps not.
If it was absolutely critical this could be done, but it would be a long project unless you are already familiar with all four of the areas above.