Success is not something you hear most Christians chant. Typically, we associate success in this world with sinful gains. "What did that person do to become successful? They probably did something sinful!" But the Bible speaks of success and biblical authors even ask God to give them success.
In Psalm 118:25, David says, "Save us, we pray, O Lord! O Lord, we pray, give us success!" Is David asking for money? A better job? Career advancement? No, he is asking for spiritual success. Specifically, he is asking for success in the Lord's provision for salvation. David knows there is a promised Messiah, and when the day comes to reveal this king, David says it will be the Lords doing (v. 23) and it will be a day for rejoicing Cv. 24).
In all that hope, David cries out to God to include him in this joyous day, and he says, "Save us!" In this request to be included in the Lord's salvation of the promised Messiah, David asks God for success. Success in being saved. Success in being chosen. Success that only the Lord can grant.
Therefore, it is not unbiblical to desire success. It is the type of success we request that matters. God wants us to desire success in spiritual matters, in salvation first, then in righteousness and holiness. It is not wrong to ask God for success in your earthly endeavors, but they must be prioritized below spiritual success in becoming more like Christ.
Meaning, it is beneficial to pray for success, and it is vital to your children that You pray for their success. First, pray that God would give your child success is knowing Jesus as their Savior. second, pray that God would give your child success in all other areas of life, such as, school, work, relationships, etc. As long as these other successes do not impede your child's success in spiritual growth, Christ-likeness, and godliness, they are worthy requests.
So, make prayer for your child's success a habit. Just make sure you are properly prioritizing which successes are more important.