To be clear, in today's Gospel Jesus is clearly teaching about intercessory and petitionary prayer. In other words, he is teaching about the kind of prayer in which you ask God for something either for yourself or someone else. Notice that it is not a question as to whether or not you should ask God for things. You should.
Of course, God doesn't always give us what we ask for. We should be grateful for this. Other times God answers our prayers in the most unexpected ways. By unexpected, I mean in a way different than you imagine he will. You see, it is not atypical to pray to God for a specific outcome and also seek to dictate to God, or at least imagine, just how you would like to see your desired outcome come to pass. In other words, very often we seek to dictate to God not only ends but means. This is very different from asking.
Persistence in prayer is not required because God isn't paying attention the first 10 times we ask. Persistence is prayer is not required because God likes us to beg for what we receive. Persistence in prayer is not required because there is a magic number of times we must ask to achieve our desired result. Persistence in prayer is not required because God doesn't know what we want and what we need.
My take on the need to be persistent in prayer is that it boils down to two things. First, persistence in prayer teaches us how to pray. We are to pray always in the will of God. Second, it teaches us what we should pray for in a given instance. Being persistent in prayer helps you determine both the form and content of a petitionary prayer.
My will is not God's will. For example, for someone who is sick, it makes sense to pray that God will heal them. Of course, God does not always do this. However, as the sacrament of anointing of the sick makes clear, taking its cue from the Gospel, especially passages like the one that was our reading last Sunday (see Luke 17:11-19), there is a healing we all need that is greater than our need for physical healing.
Sometimes it is God's will to heal a person. Other times it is not. Sometimes healing is realized through the course of medical treatment. Other times it happens suddenly and miraculously. I was fascinated to read about the second miracle that occurred through the intercession of Saint John Henry Newman, the miracle that allowed him to be made a saint:
Melissa Villalobos, a Northwestern Law School graduate who, in 2013, was pregnant with her fourth child when she began bleeding from what doctors determined was a blood clot wedged between the placenta and the uterine wall. This condition imperiled the life of both mother and unborn child, and her doctors had no medical or surgical solution. At one point, when Villalobos found herself lying in a widening pool of blood on her bathroom floor, she prayed to Newman, to whom she had developed a strong devotion. Within minutes the hemorrhaging stopped. Days later her doctors told her that she and her child were medically safe—and that they could not explain the sudden and complete healing. Neither could the teams of doctors who examined her case for the church’s canon lawyers (see "An Improbable Saint")One of my favorite Gospel passages is about the paralytic man who, waiting outside the house where Jesus was healing, had his friends lift his stretcher to the roof and then lower him down in front of the Lord. Seeing him, the Lord tells him, "Child, your sins are forgiven." This is the healing the man really needed. But grasping that some in the room thought him a blasphemer for claiming to forgive sins, Jesus physically healed the man, saying- "But that you [the ones doubting he could forgive sins] may know that the Son of Man has authority to forgive sins on earth... I say to you [the paralytic], rise, pick up your mat, and go home" (see Mark 2:1-12).
Whenever someone asks me to pray for for them or for someone else I want to know the name of the person for whom I am praying. I also want to know some particulars of the situation. I need context to be able to pray for that person with any efficacy. It's only praying for someone over time, staying in contact with them, even through a proxy, that I come to know how should pray for them and for what I am praying with some degree of specificity. So, I check with people on the condition of the one for whom I have agreed to pray.
During Lent of 2018, in one of his homilies for daily Mass, Pope Francis preached on how to pray for someone. His words are truly those of someone who is speaking from experience and not merely the postulation of a theory (see "Pope at Mass: 'Prayer requires courage and patience'"). The Holy Father spoke of the need to pray for what you want from your heart. Pull no punches. Tell God what you want and why you want it. Next, you must be courageous and patient in your prayer. Finally, your heart must be concerned for the person or petition you bring before God: "If I want the Lord to listen to my requests, I must return, and return again, to knock at the door of God’s heart, since my own heart is committed to [this petition]! But if my heart is not concerned for this need, or the person for whom I am praying, neither will it be capable of courage and patience."
It is a commitment to agree to pray for someone. Unless you're willing to commit, then don't agree to do it. As Pope Francis said in his homily: "We cannot promise someone we will pray for them, pray only an Our Father and a Hail Mary, and then leave it at that. No. If you agree to pray for someone else, you must take this [other] path. And patience is needed."
You should agree to pray for others whenever you can. In fact, you should always be praying for someone. Fasting can also help focus in on what you should pray for on someone else's behalf.
There are a few petitions for specific people I have prayed lifted to God daily for years. If nothing else, it is by praying constantly and persistently to God, our loving Father, our just and merciful judge, that we ensure there is faith on earth.