What are $H(i\omega)$, $H(s)$ and $H(z)$ called?
I would consider them to all be 'transfer' or 'system' functions. 
Those two names are interchangable, but in control theory books I see then listed more as transfer functions rather than system functions.

The domain you are in, whether it be fourier, laplace, Z or time, isn't entirely relevant to the name in that respect.