An excellent way to pray for someone is to pray a short prayer for them, then say the Jesus prayer for the person, for a set number of times.
One may say "Lord Jesus Christ, have mercy on N", or, more inclusively, "on us".
After the Jesus prayers, one may repeat the short prayer, or use another one. Two suggested prayers are below.
Prayer in Way of the Pilgrim, for others:
Merciful Lord, may Thy will be done;
Thou desirest that all men come to the truth and be saved, have mercy and save Thy servant N.
Receive this petition from me as a cry of love which Thou hast commanded
To pray for someone more diligently:
Save, O Lord, and have mercy on Thy servant(s) ________, [bow]
Deliver him (her, them) from every tribulation, wrath and need [bow],
From every sickness of soul and body, [bow]
Forgive him (her, them) every transgression, voluntary and involuntary, [bow]
And do whatever is profitable for our souls. [bow].
From the Old Believer Prayer Book.