We use the TNG100 simulation of the IllustrisTNG project to investigate the stellar specific angular momenta (j*) of ~12 000 central galaxies at z = 0 in a full cosmological context, with stellar masses (M*) ranging from 109 to $10^{12} \, {\rm M}_{\odot }$. We find that the j*-M* relations for early-type and late-type galaxies in IllustrisTNG are in good overall agreement with observations, and that these galaxy types typically 'retain' ~10-20 and ~50-60 per cent of their host haloes' specific angular momenta, respectively, with some dependence on the methodology used to measure galaxy morphology. We present results for kinematic as well as visual-like morphological measurements of the simulated galaxies. Next, we explore the scatter in the j*-M* relation with respect to the spin of the dark matter halo and the mass of the supermassive black hole (BH) at the galactic centre. We find that galaxies residing in faster spinning haloes, as well as those hosting less massive BHs, tend to have a higher specific angular momentum. We also find that, at fixed galaxy or halo mass, halo spin and BH mass are anticorrelated with each other, probably as a consequence of more efficient gas flow towards the galactic centre in slowly rotating systems. Finally, we show that halo spin plays an important role in determining galaxy sizes - larger discs form at the centres of faster rotating haloes - although the trend breaks down for massive galaxies with $M_{\ast } \gtrsim 10^{11} \, {\rm M}_{\odot }$, roughly the mass scale at which a galaxy's stellar mass becomes dominated by accreted stars.