The two low-energy isomers of C61H2 are the parent molecules of fulleroids and methanofullerenes. We present a thorough study of their structural and electronic properties within the density functional theory, in the Ideal-density approximation and also including gradient corrections to the exchange and correlation functionals. Calculations are performed both on the isolated molecules and on the solid phases. Changes with respect to C-60 as well as differences between the two C61H2 isomers are described with regard to geometrical characteristics bond patterns, valence charge distribution, and electronic energy bands. Our results can be considered a paradigm for the study of fulleroids and methanofullerenes.