I’m sorry to hear of the difficult situation. Although we virtually never pray that a person die, you are right that if such a person is suffering we should not simply pray that his life be prolonged. The ideal is to pray that the sickness leave the person. Although on the surface this is a request that the person should recover, God knows how this can best be fulfilled – whether that the illness leave the person or that the person leave the illness.

When Rabbi Shmuel Rozovsky of blessed memory (1913-1979, past rosh yeshiva of the Ponevezh Yeshiva) was dying from cancer, he would ask his visitors to pray that either the pain should leave him or he should leave the pain.

Thus, you should continue to pray and recite Psalms for the person, but bearing in mind the above. (See this past response on a recommended list of Psalms.)

I should add that all the above pertains to praying for the patient. In terms of care, we are obligated to do everything possible to care for the patient and relieve him of his pain. (At times there is room to cease providing life-support for a terminally-ill patient, but that is beyond the scope of this response. Each case must be discussed in detail with a competent Orthodox rabbi.) Although the family may ask what value is there to such a life, we may never make such a judgment ourselves. Who knows how much value there is in the Hereafter when a person endures additional suffering in this world? Thus, we leave such matters solely in God’s capable hands.

You have our prayers that God help your friend.

(Sources: Talmud Nedarim 40a, Ran s.v. “Ain” there, Igrot Moshe C.M. II 73:1, Teshuvot V'Hanhagot II 82.)