Is it permissible to make the intention of performing a second 'Umrah with the intention of "Here I am, O Allah, for an 'Umrah on behalf of a servant of Yours in the Fire who is in need of the reward of an 'Umrah to enter Paradise"?

An 'Umrah for an unspecified person is not valid. If the pilgrim intends the 'Umrah on behalf of an unknown person, the 'Umrah will be counted for himself. However, a person can perform 'Umrah for himself and then dedicate the reward of the 'Umrah to a specific or unspecified Muslim.