Under Stone-duality, the lattices which are isomorphic to the lattices of filters on a Boolean algebra are precisely the compact zero-dimensional frames. The lattices of filters on a complete Boolean algebra correspond to the compact zero-dimensional extremally disconnected frames. Therefore, by this answer, the lattices of filters on some set $X$ are precisely the compact zero-dimensional extremally disconnected frames where the isolated points in your frame form a dense subspace of the dual space in your frame. Translating this idea to a purely point-free context, the posets isomorphic to lattices of filters on a set are precisely the atomic compact zero-dimensional extremally disconnected frames.

Under Stone-duality, the lattices which are isomorphic to the lattices of filters on a Boolean algebra are precisely the compact zero-dimensional frames. The lattices of filters on a complete Boolean algebra correspond to the compact zero-dimensional extremally disconnected frames. Therefore, by this answer, the lattices of filters on some set $X$ are precisely the compact zero-dimensional extremally disconnected frames where the isolated points in your frame form a dense subspace of the dual space in your frame. Translating this idea to a purely point-free context, the posets isomorphic to lattices of filters on a set are precisely the atomic compact zero-dimensional extremally disconnected frames.

Under Stone-duality, the lattices which are isomorphic to the lattices of filters on a Boolean algebra are precisely the compact zero-dimensional frames. The lattices of filters on a complete Boolean algebra correspond to the compact zero-dimensional extremally disconnected frames. Therefore, by this answer, the lattices of filters on some set $X$ are precisely the compact zero-dimensional extremally disconnected frames where the isolated points in your frame form a dense subspace of the dual space in your frame. I will probably transfer this condition to a point-free context tomorrow when I am not so busy.

Under Stone-duality, the lattices which are isomorphic to the lattices of filters on a Boolean algebra are precisely the compact zero-dimensional frames. The lattices of filters on a complete Boolean algebra correspond to the compact zero-dimensional extremally disconnected frames. Therefore, by this answer, the lattices of filters on some set $X$ are precisely the compact zero-dimensional extremally disconnected frames where the isolated points in your frame form a dense subspace of the dual space in your frame. Translating this idea to a purely point-free context, the posets isomorphic to lattices of filters on a set are precisely the atomic compact zero-dimensional extremally disconnected frames.