What is the best kind of petitionary prayer (du'a) of a believer for another?

It is one in accordance with the conditions of its acceptability. A prayer is more or less acceptable in consideration of certain conditions being met.

For example, when one prays to God for something, he should first cleanse himself by asking for forgiveness from God, and then call God’s blessing on the Prophet Muhammad as an intercessor before and after the prayer. For calling God’s blessing on the Prophet is an acceptable prayer and the prayer said between two acceptable prayers is usually acceptable also. In addition, such a prayer should be said in the absence of the believer for whom it is said and be the kind of the prayers mentioned in the Qur’an and Hadith. For example, one should prefer comprehensive prayers such as: O God, I ask forgiveness of You, for me and him, and soundness in religion, in this world and in the Hereafter! Our Lord, grant us in the world good, and in the Hereafter good, and guard us against the chastisement of the Fire!

One should pray with sincerity, from the heart and with a religious seriousness and solemn reverence. Also, one should do so after the five daily prayers (salat) and, particularly, after the dawn prayer and in blessed times such as Friday – especially during the hour when prayer is absolutely accepted – the three months of additional prayer before Ramadan, during Ramadan itself, and, most particularly, the Night of Power. Further, one should make one’s petition in the mosque. God is expected, through His Mercy, to accept a petition made in observance of such conditions, and He accepts it so that either it is answered in this world or the one in whose name it is made will hopefully benefit from it in the Hereafter. For this reason, if one does not obtain the result for which he prayed, he should not think that the prayer has not been accepted, rather, he should consider that it has received a better acceptance.