The Pali word that's translated "lion's roar" is sīhanāda.
This dictionary lists a few Pali suttas where it's found, and gives a definition as,
sīhanāda : (m.) lion's roar; a brave speech.
I think's it meant to imply unafraid or confident, and not just loud.
Google suggests it's used in Tibetan teaching too, in a phrase or doctrine related to ...