After installing the 9.3 version of ArcGIS License Manager and restarting the computer, a black command prompt window appears that reads "ARCGIS: FLEXnet vendor daemon" across the top. Closing this box causes the ArcGIS License Manager service to stop.
The ArcGIS License Manager service is acting as though the 'Allow service to interact with Desktop' option is enabled, whether it is or not.
Solution or Workaround
Navigate to Start > Settings > Control Panel > Administrative Tools > Services.
Right-click on the ArcGIS License Manager service, select Properties and go to the Log On tab.
There is an option there that says 'Allow service to interact with Desktop'. If it is checked, uncheck it, and save that setting. If it is not checked, check it, save the setting, uncheck it again, and save that setting.
Once the change has been made, restart the ArcGIS License Manager service by right-clicking on the service and selecting Restart.