It depends. With a PassMe, depending on which one you find and purchase, you can use just about any type of computer memory card (SecureDigital, CompactFlash, Memory Stick, et cetera). The least expensive of the bunch is CF, but SD has a higher capacity.
I've also heard that the new or a new DS coming out (unsure if it's the Lite or not) will be ready to play MP3s and movies straight out of the box, which means that empty DS and/or GBA cartridges might be sold for this purpose.
A lot of people forget that there's a small amount of space on either the PSP or DS that would allow for alternate OS to operate them. The PassMe, required for the DS to load LinuxDS and play MP3 and movies, costs anywhere from $50 to $70, after shipping. The PSP can use Firewire, so no additional cost there :rolleyes unless you break the one which comes with the PSP (one does come, right?). If you can find one, the DS ends up costing $180 to $200 or $250 after geting a 1Gb memory card.
There I've wasted a lot of time just talking about how much a PassMe will add to the cost of the DS. Of course, the PSP does have the advantage of not requiring an external device, but the DS still comes out a a lower cost with everything.