Alright, I had to uninstall the Intel Rapid Storage Technology user interface from my computer because it was starting a Verify & Repair after every forced restart of my computer (it's been crashing a LOT).
After that, I knew I needed to get this fixed quick, so I went to the download page:
https://downloadcenter.intel.com/product/55005/Intel-Rapid-Storage-Technology-Intel-RST-
And I needed to find out the current version of my RAID driver, so I went to Administrative Tools / Device Manager / Disk drives / Right-clicked my RAID array / Properties / Driver tab / Driver Details / Click C:\Windows\system32\DRIVERS\iaStorF.sys.
That let me know that I had version 13.2.4.1000. I downloaded the setup for that version and reinstalled it. This did not fix the system tray icon. However, I decided to reinstall again, and selected "Repairs errors in the most recent installation by fixing missing and corrupt files, shortcuts, and registry entries."
That finally fixed it and I have the system tray icon back.