Say we have a random variable $X$, and a function $g(X)$.
I'm looking at expressions of the form $$ g^{-1}\left( E( g(X) )\right)$$ where $E$ is the expectation operator.
It's a generalization of the $p$-norm, but is there a more-or-less standard name for that sort of thing?
Your question contains very little information (about what $g$ is,... ),so it quite hard for us to give you a satisfying answer.
You might be interested in Orlicz spaces, and thus Orlicz norms. They can be applied to random variables (see here) and they generalize the $L^p$ norm.