Will the mother be sinful if she lends her son money to buy a router, knowing the possibility of it being used for disobedience, even though she stipulated that he not watch anything that angers Allah, and he fears that his resolve might falter and he might break his promise?

If a mother stipulates to her son that he should not watch anything that displeases Allah, and he covenants with her to that effect, then there is no harm in her buying him the device. This engagement might even be more beneficial for him, especially with the existence of the covenant, which helps her monitor him and remind him of Allah.