The Right and Authority to become the Children of God – to be called Christians

But as many as received him, to them gave he power¹ to become the sons of God, even to them that believe² on his name: which were born, not of blood, nor of the will of the flesh, nor of the will of man, but of God.

‭‭John‬ ‭1‬:‭12‬-‭13‬
power¹ — the "right"         
       — or rightful claim,         
       — the inward capacity,         
       — the title,         
       — warrant,        
       — authority,  

It is not merely possibility or ability, but legitimate right derived from a competent source - the Word. 

It is also "privilege", as εξουσιαν implies; to become the sons of God — To stand related to him, not merely as subjects to their king, or servants to their master, but as children to their father; being taken under his peculiar protection, direction, and care; being favoured with liberty of access to him, and intercourse with him, and constituted his heirs, and joint heirs with Christ of the heavenly inheritance.
believe²

The present participle, believing, indicates the present and continuous activity of faith. 

The word is used by John, sometimes with the dative case simply meaning to believe a person or thing; i.e., to believe that they are true or speak the truth. Thus, to believe the Scripture (John 2:22); believe me (John 4:21); believe Moses, his writings, my words (John 4:46). 

At other times with a preposition, εἰς, into, which is rendered believe in, or believe on. So here, John 6:29; John 8:30; 1 John 5:10. See the two contrasted in John 6:29, John 6:30; John 8:30, John 8:31; 1 John 5:10. 

To believe in, or on, is more than mere acceptance of a statement. It is so to accept a statement or a person as to rest upon them, to trust them practically; to draw upon and avail one's self of all that is offered to him in them. Hence to believe on the Lord Jesus Christ is not merely to believe the facts of His historic life or of His saving energy as facts, but to accept Him as Savior, Teacher, Sympathizer, Judge; to rest the soul upon Him for present and future salvation, and to accept and adopt His precepts and example as binding upon the life.