Is there a systematic theory for Gibbs measures (better if on Hilbert spaces)?

Any probability measure $\mu_1$ absolutely continuous with respect to $\mu_1$ can be written as a Gibbs measure if you allow $G$ to take values $\pm \infty$. If the density is bounded above and below, $G$ will be bounded. So you're basically asking about how to sample from a probability measure. This is a big field of study.
Markov chain Monte Carlo (MCMC) methods are commonly used, but they may run into difficulties, especially when the energy landscape has deep valleys separated by high barriers.

The field of improving convergence of sample averages is known as "enhanced sampling". As Robert pointed out, this is an incredibly hard problem. In my field (theoretical chemistry), we have been struggling with it for the last-half century.

The correct way to approach this problem on depends heavily on what information you have available to you. The simplest case is when you can evaluate $Z$ exactly. In this case, your best bet is likely survey sampling. Failing that, let's assume you can evaluate $G$ at all points $x$ at reasonable cost. If:

1. you have a good idea of what $\mu$ looks like "globally" (i.e. you know where in $H$ its modes are located, their variances, and the decay of the tails as you move away from $G$) your best is likely to be importance sampling.
2. you don't know what $\mu$ looks like, but you know it is dominated by movement along a known lower-dimensional manifold, you can try umbrella sampling or metadynamics.
3. you can evaluate $G$ at all points $x$ at reasonable cost, and you know that $G$ increases reasonably gently, you can try parallel tempering (this is basically the same as umbrella sampling, mathematically, just applied in a different way).
4. you can evaluate $G$ at all points $X$, and you have samples from a related distribution $\pi$ that you don't understand well, but is easy to sample, you can try annealed importance sampling or Hamiltonian replica exchange.
5. evaluation of $G$ is possible but expensive, and the problem isn't too high-dimensional, you can look into surrogate modeling.
6. you can't directly evaluate $G$, but you have a dynamics that preserves $\mu$, you can try Nonequilibrium Umbrella Sampling.

There are many other options and algorithms and this a very much an active area of research. However, hopefully this is enough to point you in the right direction. If you are looking for a mathematical treatment of the subject, "Free Energy Computations: A Mathematical Perspective" could be a good start. It's a bit hard to recommend reading without knowing more about the type of problem you have at hand. Nevertheless, this should hopefully be a good start.