When name is used in this way, it does not mean "the word by which a person is called," but rather "the whole nature or character of the person" as far as we know it or understand it. This is why the proverb says, "A good name is to be chosen rather than great riches" (Proverbs 22:1; see Ecclesiastes 7:1).
"Our help is in the name of the LORD" (Psalm 124:8). David, the psalmist here, does not mean that our help lies in the fact that God is called Yahweh. This name contains nothing magic in it at all. David means that our help is in His love, in His mercy, in His power, in His promises. It is in the things that have been revealed to us as parts of His nature, of His attributes, and of His character—and whether or not we have faith in them.
So then, to "believe in the name of Jesus Christ" means to believe in the nature and the character of Jesus Christ. It means to believe that He is the Son of God and that He stands in relation to the Father in a way that no other person in the universe ever has or ever will. It means that He can perfectly reveal the Father to us. It means that we believe He is the Savior, the High Priest, the Mediator and Intercessor, and our soon-coming King. It means that we believe that through Him we have entrance into God's presence—not just entrance to Him, but actually, fellowship with Him.