Does hyperconjugation into sigma-star explain methylamine's increased basicity relative to ammonia?
Yes, I once thought about this interesting phenomenon also. There is no good reason why hyperconjugative interactions cannot take place between the $\ce {N}$ lone pair orbital and the $\ce {C-H}$ $\sigma$. However, the interaction may be less favoured if you consider the geometry, compared to the the usual hyperconjugation to alkene $\pi$ systems or to carbocation $\ce {p}$ orbitals. Although geometry does not help to favour this interaction as much, the change of atom from $\ce {C}$ in the usual case to $\ce {N}$ does help to decrease the energy gap for the interaction since $\ce {N}$ is more electronegative than $\ce {C}$ and thus, its atomic orbitals would all be of a lower energy compared to $\ce {C}$. 

Raising the energy of the HOMO of a general nucleophile would increase its nucleophilicity by decreasing the energy difference between its HOMO and the LUMO of the electrophile. In the same way, by this filled orbital-filled orbital interaction, the HOMO of the methylamine is now raised, allowing it to more easily interact with acids. Thus, $\ce {MO}$ theory can be used to explain the increasing basicity of amines with increasing alkyl substitution.