A few of the many triply periodic minimal surfaces can be generated from a quadrilateral slice through a tetrahedron of mirrors, as refined by Surface Evolver. I had the idea that the same concept, applied to one of the analogous tetrahedra that tile spherical 3-space, could result in a pretty model.
To my disappointment, none of the non-prismatic kaleidoscopes — those that generate the six regular polychora — hosts a nondegenerate minimal surface (of this simple form); but each of the duoprism kaleidoscopes gives at least one.
(2018) One could try minimal surfaces with the constraint of partitioning into equal volumes, if one understood how to specify such a constraint in non-Euclidean space. Apparently it needs a metric tensor, whatever that is.