Flash memory or USB memory is similar to disk drives. In fact it is recommended you 'reformat' them often in order to keep the entire memory clean and workable. I would recommend you reformat your 256MB flash memory with the software that came with it. If none came with it, do a web search on the manufacturer type. There you will find free downloadable software to use to accomplish this task.
Remember that USB flash memory in a system is handled like another disk drive. Therefore you must contact the manufacturer determine if the flash memory can be 'reformatted' to meet the new requirements. The flash memory I have can be done like this so it allows a different format.
The 4GB SanDisk Compact Flash card features an advanced design from SanDisk that allows it to operate in cameras that use either the FAT16 or FAT32 file formats. It is the only Compact Flash card of its capacity and compatibility level that is available in the popular Type I format and fits into any Compact Flash slot.
The new 4GB SanDisk Compact Flash card features a three-position switch located in the left-hand area of the card so consumers can switch between either a single 4GB (FAT32) volume or two separate 2GB (FAT16) volumes. The switchable 4GB card ensures that users of cameras with either file format will be able to use the new card. For maximum customer satisfaction, SanDisk will initially ship the card in a FAT16 configuration-two FAT 16 partitions-eliminating the need for most users to reposition the switch and reformat.
As you mentioned, the briefcase can be trickier and to be perfectly honest I would not use it myself. I prefer to simply copy the data I want backed up directly on to the device. Have you tried doing this and then "testing" to see if it's still there after coming out of hibernation? Copy and paste a few files onto the device. Close the lid of your laptop and wait for a bit. Open the lid and after the pc comes out of hibernation, see if your files are there.
Hopefully that will work and if so I'd stick with that option. Other than that you could try contacting the manufacturer or having a look on their website, it could be a known issue which hopefully has a fix?
Flash drives usually work with Windows XP right out of the box. For Windows 98, you usually need to install drivers for the particular drive that you have. If you go to the flash drive manufacturer's web site, they will more than likely have drivers for Win.98. Well, if your flash drive is still functioning fine then it would be the motherboard or even the port coming from the motherboard or card, depending on the location of the USB port.
The USB drive could cause an issue but it is a minor device. To be honest, I don't know if I could determine that without being there to check the devices even if you gave me more details. My first gut is the system board or PCI card that is hosting the USB port is bad. Sometimes the actual port will just die but the others on the PC will work. The main issue is that these USB ports have a small amount of power going through them to power certain USB devices and a cheap USB drive might get fried.