Well, when I pray for someone, I pray that God will be patient and longsuffering with them, and not "give them over" to a reprobate mind as shown in Romans. I pray that God would bring about those situations that might influence them to be saved.
I believe in free will, but I do not understand this to be free of influence. It is impossible to be free of influence. But free will is that ability to make our own choice between two or more options.
Praying for someone's salvation is similar to when you go on visitation for someone who used to be a regular member, but has stopped coming to church. You go over and try to find out why they have not been coming. It might be the church upset them, but oftentimes it is because of some sin in that person's life. Whatever, you try to remind them why they need to be in church, that it is pleasing to God and his commandment, but also the benefits to them and their family. It can simply be that everyone in the church loves them and misses them and hopes they will come back. This can be very effective. So, this is influence and persuasion, but the person has to make their own decision.
There are so many various ways you can influence a person to do anything. You can lay the ol' guilt trip on them. Or you can use a positive argument, that their kids will grow up to love God and be better people if they attend church, etc...
So, this is what I pray, that God would not give up on this person and continue to draw and influence them to be saved. But the person has to make their own choice in the end.
Personally, I am glad I have free will and make my own choices, but I am often made miserable by the choices I have made. I can recall many stupid decisions I have made in life, but they were my decisions and I must take full responsibility for my choices.