If the person you're praying for is not one of the elect, why pray for them?
The way I see it is this: It isn't if they are elect or not but what we think that persons standing is. If we have reason to think the person might be a stray sheep then we should pray for them even if we are wrong. Praying for God's enemies is wrong. We must discern as far as possible who to pray for and who not to from the persons concerned.
Will God change His mind and extend grace to them because you asked Him to?
No, but He does work His works through us. He involves us. We should be led by the Spirit in prayer or we should be general in praying for the things we know He wants us to have such as knowledge and Wisdom and so on. If you find yourself praying for more love more hospitality and so on then you can be sure you will get it because it was God that put the prayer into your mind so that He could bless you and get closer to you.
Please don't say because God says to pray for them because you don't know who the elect are.
We should be led by the Spirit in prayer. God does tell us who to pray for but being told by God does not mean that we can know we are praying for the elect. I can think of reasons why God gets us to pray for reprobates, it could be God wants them shamed on Judgment Day for their thoughts of us.
RO 8:26 In the same way, the Spirit helps us in our weakness.
We do not know what we ought to pray for, but the Spirit himself intercedes for us with groans that words cannot express. 27 And he who searches our hearts knows the mind of the Spirit, because the Spirit intercedes for the saints in accordance with God's will.
We don't need to get it right the Holy Spirit prays on our behalf and gets it right. We should use our prayers for selfish reasons. Ask for more and more of Him so that we become the people our Father will be proud of like He was of Job.
I do not understand Cals praying for the salvation of someone. It makes no sense to me.
It makes us feel better.

How do you know, wife, whether you will save your husband? Or, how do you know, husband, whether you will save your wife? 1 Cor 7:16.
Praying or not doesn't change God's plan. The plan is being worked out. Unbelieving wifes and husbands are made holy and are in a priveledged position to hear and see the word of God in action. Since the non-Christian is loved then that love will express itself in pleas of mercy for the husband or wife of the Christian with a half promise,
how do you know?. Love demands our prayers. Love is kind and wants the best for the objects of our love.
Why pray to God if the final decision is up to man? It doesn’t make sense. You asking God to do something the Arminian tells us He can’t do…make the decision to believe.
I will be interested in the response.
You could pray something like this: 'Lord, I pray for my friend _____ to be saved, but if he is not one of the elect then I pray you send him to hell for all eternity.'
That way you've got it covered and you will be praying God's will.
We do not need to give Him options, Jesus said, "Your will be done."
I think that's covered with "not my will, but thine" isn't it? Absolutely.
john.