The balancing properties of a gömböc are affected by mechanical defects and dust both on its body and on the surface on which it rests. If damaged, the process of restoring the original shape is more complex than producing a new one.
The self-righting properties degrade in imprecise environments -- from the level of "100% mathematical certainty it can right itself" -- to something sort of absolute mathematical certainty. Not from self-righting to complete failure.
Dropping an eyelash on it, and using a standard kitchen table probably won't make it fail, and it will probably still work -- but it does impact it at the theoretical absolute level.
I bet if that eyelash were on the stable equilibrium point, you'd now have two stable equilibrium points (one on either side of the eyelash, basically), and possibly an additional unstable point (balanced on the eyelash).
The self-righting property would be completely unaffected if the surface imperfection is smaller than a certain threshold value. This value is a function of the "equilibrium-ity" for any point of the surface.
Close to the stable and unstable equilibrium points, any imperfections create additional equilibrium points, like someone else mentioned.
The threshold will grow with the distance to equilibrium points, so in some places, an eye lash, a grain of sand or the inscribed logo makes absolutely no difference.
The balancing properties of a gömböc are affected by mechanical defects and dust both on its body and on the surface on which it rests. If damaged, the process of restoring the original shape is more complex than producing a new one.