Is it wrong to pray to God and ask Him to bless atheists if they are going through tough times in life and to give them the strength and courage to continue to face whatever struggles they may be facing?

Consider this divine directive given to Christians:

1 Timothy 2:1-4 (ESV)
¹ First of all, then, I urge that supplications, prayers, intercessions, and thanksgivings be made for all people,
² for kings and all who are in high positions, that we may lead a peaceful and quiet life, godly and dignified in every way.
³ This is good, and it is pleasing in the sight of God our Savior,
⁴ who desires all people to be saved and to come to the knowledge of the truth.

First, take note that God directed that “supplications, prayers, intercessions, and thanksgivings be made” specifically for “all people” (v. 1).  That would include “kings and all who are in high positions” (v. 2), including the rulers of Paul’s time who did not believe in or even know of the one true God and who were polytheistic in their own religious beliefs, many of whom even going so far as to declare themselves to be gods.  Therefore, from this passage alone we can correctly conclude that God would want us to pray for atheists.

This would include asking God to intercede on their behalf (the meaning of “intercessions”).  After all, God “makes his sun rise on the evil and on the good, and sends rain on the just and on the unjust” (Matt. 5:45).  He created and cares for atheists just as He created and cares for believers.  Even though He gives more blessings to those who follow Him (cf. Eph. 1:3ff), He certainly also blesses those outside of Christ in certain ways, including atheists.  So we should pray for them.

This would even include militant atheists who are adamant, purposeful opponents of Christianity like Richard Dawkins.  Remember Jesus’ directive in the Sermon on the Mount:

Matthew 5:43-48 (ESV)
⁴³ “You have heard that it was said, ‘You shall love your neighbor and hate your enemy.’
⁴⁴ But I say to you, Love your enemies and pray for those who persecute you,
⁴⁵ so that you may be sons of your Father who is in heaven. For he makes his sun rise on the evil and on the good, and sends rain on the just and on the unjust.
⁴⁶ For if you love those who love you, what reward do you have? Do not even the tax collectors do the same?
⁴⁷ And if you greet only your brothers, what more are you doing than others? Do not even the Gentiles do the same?
⁴⁸ You therefore must be perfect, as your heavenly Father is perfect.

One last point.  It is worth considering the reason God through Paul directed us to pray for “all people” in 1 Timothy 2:1-4:  “This is good, and it is pleasing in the sight of God our Savior, who desires all people to be saved and to come to the knowledge of the truth” (vs. 3-4).  If we sincerely pray for the best for atheists, we will be praying that they come to know and accept the truth and be saved.  If we sincerely pray these things for them, we will also be letting our light shine around them, including what would benefit them in the good works we do, which could result in them glorifying God because it might just open their minds to be receptive to the gospel message God wants us to share with them (Matt. 5:16; Mark 16:15).