This may be silly to ask. Consider the internal circuit board of any electronic device (cell phone, for example) driven by a micro-controller, also on the same board. I'm about to write a firmware that will have features to test various peripherals and other things on the board (IO pins, EEPROM, LCD etc).
Is there any common name used for such firmware?
For example, a bootloader is a kind of firmware that allow the code memory to be written during run time.
It will not be self test. External commands will be given to the micro and the micro will execute them. Each command will test a specific feature.