Jesus Christ refers to God's name three times in this brief and exceedingly important prayer! The name represents what He spent His ministry revealing to us about God. He keeps us through His name both by our trusting in what it means and our obedience to how it shows we should live.
He defines eternal life as "to know God." "Know" suggests a very close intimacy, just as a husband and wife are intimate in marriage (Genesis 4:1). It indicates experiential knowledge, not theoretical. In Amos 5:4, God exclaims, "Seek Me and live!" He is saying, "Turn to Me and My way of life; seek to know Me," not "Search for Me," because He has already revealed Himself to us. He is saying, "Seek to know Me by living the same way I do." That is how experiential knowledge of Him becomes an intimate knowing of Him. He will walk with such people (Amos 3:3).
Aionis, the word translated "eternal," deals less with duration of life (although it is included), than it does with quality of life. Living endlessly is not necessarily good. Would anyone want to live forever with a demon's quality of life? True eternal life is the life of God. To possess it means experiencing now some of its splendor because it is being lived and producing its glorious fruits.
Psalm 9:10 adds, "And those who know Your name will put their trust in You; for You, Lord, have not forsaken those who seek You." Those living by faith do not trust in what He is called, for that would be mere superstition. Their faith is in what He is, His character and nature, which they have experienced by seeking to live His way.