Commentators explain that when David prays to be saved "by thy name," he is appealing to the fullness of God's character. It's a plea based on everything God is: His power, faithfulness, goodness, and reputation. As John Gill notes, it can also mean praying "for your name's sake," asking God to act for His own glory. This teaches believers to base their prayers not on their own merit, but on God's revealed character.