The question is a fair question to ask. But often the dogma that prompts the question is based in a hyper-Calvinist assumption that presumes that the answer must be that God does all the choosing and therefore we should not be so arrogant as to impose our will or desires upon God.
It is true, of course, that God does all the choosing (I will neither defend nor apologize for that doctrine here). But God's choosing does not militate against our responsibility to act and our culpabiliity for our choices. Moreover, many times God tells us to pray for our desires (like seeing others saved) while always adding this contingency, "If you, the Lord, also will it" (see James 4:15).
In fact, God will accomplish all He desires to do no matter what we do. And yet, He also tells us that He will withhold certain blessings and certain answers to prayer if we decide not to pray (see James 4:2), and that He will hold us guilty if we do not preach righteousness to the unsaved (see Ezekiel 3:18, 33:8 ) even though it is He who decides who will be saved. Both are true, His choosing and our responsibility to act.
Paul prayed for the believers in Ephesus, giving thanks to God for their faithfulness (see Ephesians 1:15-17). But Paul also prayed for the unbelievers who were in the church at Ephesus, begging God to enlighten them to the inheritance that only belongs to those of us who believe.
I pray that the eyes of your heart may be enlightened, so that you will know what is the hope of His calling, what are the riches of the glory of His inheritance in the saints, and what is the surpassing greatness of His power toward us who believe. These are in accordance with the working of the strength of His might ... (Ephesians 1:18-19)
While some may argue Paul is not here praying for the unsaved, I would ask why he would pray for englightenment for those already enlightened; why would he pray that those who already know the hope of Christ's calling would yet come to know that hope; and why those who have already laid claim to the riches of the saints' inheritance would need to yet come to lay claim again to that inheritance which they already claim?
Why would we not pray that the unsaved would repent and become saved? Is it not God's will that all become saved; so why would we not pray that God's will be done on earth?
Finally, we have this commandment from Paul to pray for the unsaved among us, that all men may be saved and come to the truth:
First of all, then, I urge that entreaties and prayers, petitions and thanksgivings, be made on behalf of all men, for kings and all who are in authority, so that we may lead a tranquil and quiet life in all godliness and dignity. This is good and acceptable in the sight of God our Savior, who desires all men to be saved and to come to the knowledge of the truth. (1 Timothy 2:1-4)
