Does hyperconjugation into sigma-star explain methylamine's increased basicity relative to ammonia? The inductive effect with the polarisation of the C-N bond obviously plays a role in stabilising the positive charge of the conjugate acid. But given the similar electronegativities for C and H, CH3 group should donate as much electron density to N as H does (roughly), if only the bonding atoms are considered. And if we're considering atoms other than the two bonding atoms in their contribution to stability, is it not a question then of hyperconjugation?

This question seems to deny hyperconjugation plays a role in stabilising ammonium derivatives. However, it takes a very narrow view of hyperconjugation and focuses only on electron density donation into bonding orbitals. But if more substituted double bonds are more stable because of the donation of alkyl groups into pi-star orbitals, antibonding orbitals must be viable recipients. And similarly in methylammonium, we have sigma-star orbitals on the positive nitrogen which the C-H bonds of the methyl should be able to donate into, destabilising the N-H bond in methylammonium but overall stabilising the system.

So, my question is does hyperconjugation from sigma C-H bonds into sigma-star orbitals of sp3 centres occur, and then from that - does hyperconjugation explain methylammonia's strength (or alkylammonias' strength, more generally) 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.