What is the expiation for a supplication made in jest, with the intention of making others feel hot, knowing that it was uttered during a time when supplications are answered on Friday, and how can good intentions be demonstrated to the other party?

Your cousin's statement that she will not forgive you if the prayer is answered is meaningless, as forgiveness takes place in this worldly life. It is not permissible for a Muslim to supplicate against another without a legitimate Sharia (definition) right. Whoever supplicates unjustly is an oppressor and aggressor, and their supplication will not be answered. Therefore, you should seek forgiveness for this sister, pray for her, and clarify that the supplication was not answered and that you did not intend to harm her.