From: Allan Wechsler <acwacw@gmail.com> I don't know whether I actually posted my conjecture a few years ago when the "hey-for-three" solution was discovered. I'm not sure how to rigorize the conjecture, but I am convinced that there is an at-least-countably-infinite family of such stable (using the word in the usual sense, pace Warren) configurations.
Basically, if you start off three bodies in a random way, then the system might spend some energy to eject one of the three, but if it doesn't, the system will eventually come very close to one of its previous configurations; if it does, then you can construct a stable solution by tiny perturbations. --You may be right, but that reasoning is not necessarily valid. Consider the geodesic flow on a negative curvature compact hyperbolic manifold. There are an infinite set of closed orbits, which is known (also seems shown basically by Wechsler's reasoning to get a nearly-perfect closed orbit then "pull the string taut") -- but every one is unstable in the sense tiny perturbation to initial conditions grows exponentially with time. In fact every geodesic trajectory is unstable. And this is of course a Hamiltonian system in the sense energy is conserved.