I'm about to get one, and it's something I'd like to know. If it doesn't, will adding the official 250GB hard drive add it?
Based on a quick search, unfortunately it looks like it won't. While it seems like it should work (due to having 4GB of internal flash memory), it doesn't probably because the original Xbox didn't know about flash memory, only its internal hard drive. Since really running original Xbox games on the 360 boots up the equivalent of an original Xbox, that's all it can write to.
Buying the 250GB hard drive (or, better yet, just buying the model that comes with the hard drive - it's cheaper that way), should do the trick and allow original Xbox games to run on the Xbox 360 (well, those that have been updated to work anyway).