Is it permissible for an aunt who is in need of healthcare and surgery to use the private health insurance of her niece, who does not use it?

It is not permissible to resort to deception to benefit from the insurance system if it restricts benefits to your sister's daughter, as this involves prohibited fraud and falsification, and a violation of the insurance terms. Allah Almighty says: "O you who have believed, fulfill [all] contracts," and the Prophet, peace and blessings be upon him, said: "Muslims are bound by their conditions." This falls under unlawfully devouring people's wealth, and Allah Almighty says: "Do not consume one another's wealth unjustly but only [in lawful] trade by mutual consent." If the insurance is commercial, then it is not permissible to subscribe to it in the first place except for those who are forced to, and it is not permissible to benefit from it except within the limits of what was paid.