"Name" here is a metonym for the person of Jesus Christ. Alternate translation: "because of me" or "because you believe in my name" or "because you believe in me" (See: [[rc://en/ta/man/jit/figs-metonymy]])
Being discouraged is spoken of as being tired. Alternate translation: "you have not become discouraged" or "you have not quit" (See: [[rc://en/ta/man/jit/figs-metaphor]])