Terrific question and post, Homer. What's God's part and what's our part? And when do we give up praying and believing for someone?
I think it's fine and biblical to pray that God would open someone's heart. Basically, as an Arminian, I feel I can pray anything except one thing: God make the decision for them. I can't pray that. I can pray "God show them it's true; show them it's desirable; show them it's reasonable; show them how merciful and good you are; show them what Jesus has done; show them that Satan wants to destroy them; give them grace to not be offended by the unknowns." But I would never, ever pray "God force them to say yes, take away their free will."
It is said the Lord opened Lydia's heart. But Calvinists often leave out the whole passage. There is more to it. What does it say immediately before?
One of those listening was a woman from the city of Thyatira named Lydia, a dealer in purple cloth. She was a worshiper of God.
So this wasn't out of the blue—Lydia had a prepared heart inclined to God. As Christ said "If anyone
wills to know God's will he will know what teaching is
of God." Even in John 6, a passage Calvinists love, there is the strong teaching that those who are "seeking" and those who are "willing" are the ones whom the Father will draw to Christ (see my study of John 6 from an Arminian perspective here:
http://theos.org/forum/viewtopic.php?f= ... ing#p66659).
There is a time that Scripture says "don't pray anymore." Once in the OT (in Jeremiah) and once in the NT (1 John 5:16). I believe that applies to someone who has been given an extraordinary amount of grace and light. I wouldn't give up praying for someone with an initial or continued rejection of a Gospel presentation (there is a valley of decision). But once you know they've made up their mind, I think that's the sin that leads to death.
God bless!