Paul’s prayer in Romans 1:9-10 contains two points of apparent tension:
(1) Paul says that in his prayers he remembers the Christians in Rome always and always he is asking God for the opportunity to go to Rome.
(2) Paul understands that one thing inhibits him: the will of God (ἐν τῷ θελήματι τοῦ θεοῦ).
Pragmatically, these few words from Paul tell us a lot about prayer. On the one hand, prayer does matter and one should continue to pray for certain things, even frequently over an extended period of time. On the other hand, God remains God, and if God wants to prevent something or allow something then that is God’s divine prerogative.
Two antithesis to Paul’s understanding of prayer seem to be that either it doesn’t matter (e.g., prayer doesn’t change God, it changes us) or that prayer is magic, controlling the divine will. Why does God answer a prayer or not answer a prayer? Why would God not answer a prayer the twelfth time it is prayed, but does answer the prayer the thirteenth time it is prayed? These questions receive few helpful answers. We can speculate, but that doesn’t help much. What Paul does seem to believe (and something we might adopt) is that God does listen, but that doesn’t mean God is our genie. Keep praying, but remember that God is God.
the archives near Emmaus 
I’ve taken the view of prayer that God invites our opinions and will act on them based on His own will after factoring in our views. He does factor our views in though, IMO.
On the “repetitive prayer” thing. Right now, I view that the same way I do us agreeing on a prayer’s content as separate folks. That just adds to it’s usefulness with God.
More expressions of reverence for the ability of God to do what we at least think is intrinsic good and the thought that we believe He wants to help us, which is honoring His character even if our views are flawed.
We are wrestling with this in our church currently, in regards to the issue of healing. We are trying to live between the tension of expectancy and surrender. That we pray in the affirmative, believing that it is God’s will to heal, but we are also realizing that the amen is the moment when that prayer is let go and placed firmly in the hands of God.