Update: Matt Domenici claims that this problem (which is probably not specific to Mac OS 8) is due to a corrupted Apple Menu Options (AMO) prefs file. To fix it, turn AMO off in the control panel, then throw away AMO prefs; next restart and turn AMO back on. "This should fix the problem. Simply throwing away the prefs without switching AMO off and restarting won't clear the problem." Others have now confirmed this.
This remaining items from this posting have been incorporated into the Troubleshooting Mac OS 8 page.