Generalized Kummer varieties constitute one of the two known infinite families of compact hyper-Kaehler varieties arising in each even complex dimension 2n\geq 4. They are constructed as Albanese fibers of moduli spaces of sheaves on an abelian surface; however, their deformation space is strictly larger than that of the underlying abelian surface--meaning that a generic deformation of a generalized Kummer variety cannot be obtained from such a construction. Remarkably, these "hidden" deformations are encoded in the deformation theory of Weil-type abelian fourfolds, as revealed by work of O'Grady and Markman. This motivates the study of moduli spaces of sheaves on Weil-type abelian fourfolds with the objective of recovering a complete family of generalized Kummer varieties. In this talk, I will describe a construction of special sheaves on Weil-type abelian fourfolds and discuss their deformation properties. This is ongoing work with Nicolas Addington.