What you are describing is an OS issue. Or possibly a DOS issue (Even though they say it no longer exists, ha). The first name is a long file name and the second is a short file name.
The short file name is used if you are using an FAT formated drive. An option is enabled in windows to use short file names only. Another possiblity is a portion of your Table on the hard drive is corrupted.
The Fat table can be changed using a utility in windows that converts FAT to NTFS or Formatting the drive, that is assuming the drive can be changed. Some drives like flash drives for a camera may be required to stay as a FAT drive for the camera to work correctly.
The option for enabling short filenames may no longer be available. Windows use to have a place to change settings when issues come up to rule out the cause. Its possible that during Windows 98 to Windows ME it got phased out. I did not see it in Windows XP but then again quite a few options are now hidden. So if you did not change it yourself odds are you can rule out this theory.
As far as a corrupt NTFS Table goes it might be as simple as using chkdsk.exe or as bad as formating the drive. Odds are you would notice this issue when saving any or all new documents and reading a directory (folder) may show jibbersh or the short filenames only.
The tell tale sign of a short file name is the length of the name which is a maximum of 8 characters, plus the period, then the 3 character extention. If the long file name exceeds the limit of the short file name, a ~ character is used just before the period and the extention is trunicated to the first three characters after the period. Also the short file name can only use alphanumeric characters.
Hopefully this helps some. I do not think there is an option in Illustrator for determining which filename type to use. As this is usually left up to the OS.
As Silkrooster said - utterly and completely an operating system issue. It's weird, though. Ever since Windows 95 the default in any version of Windows is to allow long file names. You have a serious problem there somewhere...