Perhaps the most commonly used norm on function space is the $L^p$ norm. Put in terms of distance between functions, you would say that $$D(f,g) = \sqrt[p]{\int |f(x)-g(x)|^p \,\mathrm{d}x}.$$ The Hellinger distance between to probability distribution functions is given by (up to a scaling) $$H(P,Q) = \sqrt{\int\left(\sqrt{\frac{\mathrm{d}\,P}{\mathrm{d}\,x}} - \sqrt{\frac{\mathrm{d}\,Q}{\mathrm{d}\,x}}\right)^2\mathrm{d}x}.$$
You could immediately generalize the Hellinger distance to read $$H^p(P,Q) = \sqrt[p]{\int\left|\sqrt[p]{\frac{\mathrm{d}\,P}{\mathrm{d}\,x}} - \sqrt[p]{\frac{\mathrm{d}\,Q}{\mathrm{d}\,x}}\right|^p\mathrm{d}x}.$$ The quantity $H_p$ is, obviously, closely related to the $L^p$ norm, but it has been modified to make it invariant under reparameterizations of the integral/functions.
Does $H^p$ have any interesting properties that set it apart from $L^p$ that have been studied? You could summarize it as the $L^p$ norm on the functions that are the $p^{\mathrm{th}}$-root of the functions, so it seems like it should qualify as a good metric, even if it fails to be a norm.