By an equidistributed countable subgroup I mean a countable subgroup (with a finite or possibly countable set of generators) that is dense in $G$ such that for any sufficiently nice function (Haar measurable or stronger, although probably no stronger than continuous almost everywhere and (locally) bounded) you have $$\int f(g)d\mu=\lim_{n\rightarrow\infty}\frac{1}{\left|W_n\right|}\sum_{g\in W_n}f(g)$$ where $W_n$ (in the finitely generated case) is the set of elements generated by products of $n$ generators (and their inverses) (in the infinitely generated case $W_n$ would be something like the set of words of length $n$ consisting of the first $n$ generators and their inverses, something to ensure $W_n$ is finite at each step).
I'm primarily interested in the finitely generated case for compact Lie groups. This is obvious for compact Abelian groups and I think it might be trivial for the infinitely generated case (just let the generators be an equidistributed sequence in $G$ or something).