It sounds like richness shares with beauty: the beholder's eye determines what is rich. While I will not attempt to give a quantitative answer, I think the following points can be considered.
The object may have many properties that are desired for study or application. An example of this is a set system {{1,2,3},{1,4,5},{1,6,7},{2,4,6},{2,5,7},{3,4,7},{3,5,6}} on seven points. It has a lot of symmetry but the structure as a system is not apparent until you give it an interpretation: finite projective plane, complement of a D-optimal design, arrangement of statistical tests. Richness of application may be meant even though richness of structure was stated.
The richness may come from using the structure to build more things. The two-element lattice generates the variety of all distributive lattices, and its equational theory is the same as the equational theory of any larger distributive lattice in the same language. So in a proper context the richness may stem from how the structure will be used in creating other structures or in determining properties of related structures.
Granted these are not literal interpretations, but I think they are in mind when someone makes a statement about richness of structure.
Gerhard "Ask Me About System Design" Paseman, 2010.02.21