Case: Reusing a case is a good bet, unless it has a backplane or is proprietary in some other way. If your old case is a Compaq, IBM, Gateway 2k, or Packard Bell you will probably need a new one.
Video card: This is a good component to save if you're trying to save money, as long as it uses the same bus.
Monitor: Almost every VGA monitor is reusable, especially if you're keeping the video card. Memory: 72-pin FPM RAM can be used in over 98% of all new motherboards. If you want to try the "SIMM stackers" for your old 30-pin SIMM's you may be in for a lot of extra installation difficulties unless you have some brand new RAM to try, too.
Software: Have a bootable floppy disk, and perhaps a hard disk on hand for the installation. Windows 95 is a good OS if want to test your PC's components in a PnP environment.
Hard disk: Almost all hard disks are reusable. If you get another hard disk, save yourself some trouble and attempt to get the same brand. I have an old Conner hard disk that can't work with other brands.
Modems, speakers, CD-ROM's, sound cards, and SCSI controllers are reusable. The major problem you could run into is bus (slot) type. VESA Local Bus slots are extremely scarce, except in 486 motherboards.
High quality brand names of [personal computer] industry leaders: 3Com, 3M, Hewlett Packard, NEC, Seagate, and U.S. Robotics.
Sony, Panasonic, AT&T, DEC, IBM, Mitsubishi are known in other industries, but I don't give them the same respect as the ones above.
Notable mentions are: Adaptec, Altec Lansing, ATI, Creative Labs, Diamond, Fujitsu, Hayes, Intel, Logitech, Motorola, TI, and Western Digital.