Minor illusion can't do this at all
First, let's ignore for the moment that the inner ear is not a hearing organ (it's nearby, but is involved in balance rather than hearing), and assume your player was thinking “inside their ear canal” and mixed that up with “inner ear.”
The spell doesn't target creatures, for starters, so it would have to be placed in a specific location. Although “the monster's inner ear (canal)” sounds like a specific location, spells also require that you can see the location and draw an unblocked line between the caster and the location (“A Clear Path to the Target”, PHB, p. 203), which you can't do with an inner ear. Since the monster's head blocks both sight and the line between the location and the caster, attempting to target something inside their head would result in the effect coming into being at the point outside their head nearest the caster — probably still pretty surprising, but not what was intended, I imagine.
At best, you can use minor illusion to make a scream erupt beside a monster's outer ear, which should be plenty useful for a player wanting to startle it. The major difference there is that minor illusion doesn't let you move the sound around, so the intended victim can just move away from it. I imagine the desire to target the “inner ear (canal)” was based on hoping it would stick to them and stay inside their ear for the whole minute, but that's not something the spell can do.