If the System Menu's IOS is corrupted, then the Wii will not show anything on the screen. This will appear identical to a System Menu brick, however, Priiloader cannot be used to fix this brick.
A broken Wifi Module will present the same symptoms, but can only be fixed by replacing the module. This extends to installing a normal System Menu IOS on a Wii Mini without a WiFi module installed.
Fixing this brick
To recover from an IOS brick, BootMii must be installed as boot2. Normal homebrew will likely not work to recover from this brick, because they depend on IOS's that are now corrupted. Instead, special tools using mini (such as RestoreMii) must be used to fix this brick.
If BootMii cannot be installed as boot2 on this Wii, then Infectus must be used to restore a NAND backup.
Cause of this brick
This brick occurs when the System Menu IOS is missing or corrupted. When this happens, boot2 cannot locate the IOS that should be launched, so it fails to boot, or boots garbage code.