Should we pray for someone's salvation?

To put in another perspective, it is traditional for Orthodox Christians to pray for the salvation of not only other living people, but also for the dead. Note that when we pray for 'salvation, visitation and continuance in all good things' as the common prayer states, we do not make a distinction between believers and non believers, and ultimately, no distinction between the living and the dead. Although there is an important point here as to why this prayer is generally first towards the living - those who are glorified do not need to be prayed for, but instead are asked to entreat God for those of us who struggle. But then what that means is those people are those who did not taste of death; they, like the Thief on the cross were 'that day with Him in Paradise' to paraphrase.

To 'undigress', also from our perspective we can take note of two stark facts:

1. God wills that all men be saved. Who can countervail his will?
2. Not all men end up being saved. We know of at least Judas, and according to our tradition, Arius as well.

Given this, we should say that our praying for the salvation of people is not asking God to save them - he already has. Instead, we are asking him to help incline their heart toward him to accept the gift he freely gives. Given this, why would you not pray for their salvation? Do you not want to live with them in Eternity? Do you not want to see them glorified and at perfect peace?

I know some people I'd rather not be around. But to not want good things for them, especially the Good Things, is certainly lacking in Charity. Let a person decide for themselves where they stand on this.

I see prayer as an act of obedience. Prayer changes our hearts to be in line with God's will as we commune with Him in it. If I pray for someone it does not mean God will force salvation upon them because that would violate their free will. God desires but does not force all to be saved. If someone's salvation depended upon my prayers then it is by me and not by Christ. If I pray for someone, and my heart is open to what the Holy Spirit would have me preach and live the Gospel before them, then the transforming power of God being demonstrated to them by my life may convince them of the truth. We are commanded to preach, teach, make disciples, forgive sins; we are to be God's hands and ambassadors in this world. If I don't pray for their salvation, but pray that God would open my mind and the minds of other saints in a way that would allow the Gospel to be preached with authoritative it will be equally effective.
Prayer for Spiritual Insight Ephesians 1: 15 This is why, since I heard about your faith in the Lord Jesus and your love for all the saints, 16 I never stop giving thanks for you as I remember you in my prayers. 17 I pray that the God of our Lord Jesus Christ, the glorious Father, would give you a spirit of wisdom and revelation in the knowledge of Him. 18 I pray that the perception of your mind may be enlightened so you may know what is the hope of His calling, what are the glorious riches of His inheritance among the saints, 19 and what is the immeasurable greatness of His power to us who believe, according to the working of His vast strength.

Romans 10:17 So then faith comes by hearing, and hearing by the word of God.

Mark 15 Then He said to them, “Go into all the world and preach the gospel to the whole creation. 16 Whoever believes and is baptized will be saved, but whoever does not believe will be condemned.

The bible is the best authority, as always, I Tim.2:1-5;

1"First of all, then, I urge that entreaties and prayers, petitions and thanksgivings, be made on behalf of all men, 2for kings and all who are in authority, so that we may lead a tranquil and quiet life in all godliness and dignity. 3This is good and acceptable in the sight of God our Savior, 4who desires all men to be saved and to come to the knowledge of the truth. 5For there is one God, and one mediator also between God and men, the man Christ Jesus, 6who gave Himself as a ransom for all, the testimony given at the proper time" 1 Timothy 2:1-5

I Tim. 4:10 - "For it is for this we labor and strive, because we have fixed our hope on the living God, who is the Savior of all men, especially of believers."

I chose these verses from I Timothy from among many such verses found in the bible (for example II Pet.3:9c - "not wishing that any should perish' etc.) to illustrate that it is God's will of desire that all men have opportunity for salvation.

I am aware that strong Calvinist interpret these and like verses to apply only to the elect. I do not believe the Greek supports the translation; ex. in Timothy as "All types or categories of men" nor in II Pet. "Not willing that any of us-ward (i.e. believers) should perish." Therefore I believe it is clearly proper to pray for the salvation of the unsaved.

I find that I pray for others salvation a lot but its not a "please Lord save this sinner" type thing as it is an active prayer. I pray that their heart may be softened so that the gospel can take root in their heart and change them. I ask if there is anything that I can do to help that process and I pray for them to find comfort in trials and to know that that comfort is coming from God.

That's my take on praying for someones salvation and I think that yes its something we all should be doing. However your question is a little broad and you are going to get several different answers depending on the denomination that is answering.

As for my reasons for answering the way I did I think this scripture can help:

(KJV) Ephesians 6:17-18 And take the helmet of salvation, and the sword of the Spirit, which is the word of God: 18) Praying always with all prayer and supplication in the Spirit, and watching thereunto with all perseverance and supplication for all saints;

Paul describes something active in taking the helmet of salvation and then in watching in prayer over all the saints.

Also as @Affable Geek said it depends on what your thought process is on salvation. Some believe that it is solely up to the Lord. Others believe that it takes faith alone, and yet others believe that it takes faith and works to obtain salvation. So whether its appropriate for you to pray for someones salvation depends on where you fall in those three main groups. (And that does not even begin to cover all the different thoughts on how one obtains or is given salvation).