God answers prayers so He will be glorified. Prayer isn't primarily to give us what we want. God cares whether we have what we need, and He wants us to be blest through our petitions. He therefore responds by pouring out blessings on those who pray. But the primary objective for all of those blessings and provisions is to put God's goodness, wisdom, power and grace on display. - John MacArthur
I think this can especially be seen when we pray for others and see our prayers answered. The glory we give to God touches us in a very personal, special way because of our personal connection to the situation. Had we not prayed and events transpired exactly the same, God will still get the glory but our personal connection and testimony would not be the same. The more we pray, the more opportunity we have to share in God's glory when He moves and acts on our prayers.
The prophet Daniel gives us an example of a prayer for the purpose of God bringing glory to Himself. The Israelites had been taken captive and Jerusalem had been destroyed. Although the events are a consequence to the sin of the Israelites themselves, Daniel pleads with God that He will once again defend and restore them. He pleads are based not only on God's mercy but also to defend His honor. The enemies of Israel were doubting that God existed or believing He was powerless. Daniel was asking God to defend, not only His people, but His very name.
This is just one example of many in Scripture of God's people seeking His glory as the primary purpose of their prayers.
When I pray, do I seek God's glory above all or do I pray for my own benefit?