Is it permissible for my friend to use a computer and the internet after her divorce and after paying her expiation for breaking an oath (feeding 30 needy persons), even though she swore to her ex-husband upon the Quran and upon the heads of her children that she would not use them, and she fears that harm may befall them?

Swearing by other than Allah is a subtle form of polytheism (shirk khafi), and the swearer must repent, feel remorse, and resolve not to return to it. No expiation for an oath (kaffarat yameen) is required for someone who swore by other than Allah and then broke that oath. The expiation for an oath is feeding ten poor persons, or clothing them, or freeing a slave. If one cannot find these, then fasting for three days. Swearing by the children's heads does not change anything of what is predestined, for everything that befalls a person is by the decree of Allah. Furthermore, the internet is a double-edged sword; if used for good, it is permissible, but if there is fear of harm from it due to indecent programs and images, then it is not permissible to use it.