The “life” referred to here has 2 attributes:

1. independence
   The life of the Father is dependent upon no other.  He is self-existent. That is why He called Himself “I am.” #Ex 3:14
   Ex 3:14  God said to Moses, “I AM WHO I AM. This is what you are to say to the Israelites: ‘I AM has sent me to you.’”
2. the source of all life
   All life takes its origin from the Father. #Ge 2:7; Ps 36:9, 66:9; Deu 30:20
   Ge 2:7  the LORD God formed the man from the dust of the ground and breathed into his nostrils the breath of life, and the man became a living being.

The Jews fully understood this, but Jesus said here that the Father had given that same quality of life to the Son.  The Son was now self-existent and the source of all other life, except that of the Father.  This is further evidence of the deity of the Son, which the Jewish leaders would find very hard to accept.

“given” = for God the Father to have given this life to the Son, the latter must not have had it in the first place.  Even though the Son is God, He is not the Father and does not (or did not at one time) possess all the attributes of the Father.  Interesting!  There is equality between the Father and the Son as to this life except that the Father has not been given it, but the Son has.

Augustine describes this life which the Father has “in himself” as one which is not borrowed and also not of what He Himself is, but “the very life is to Him His very self.”  He uses the illustration of light.  Men require light to be able to see in the darkness, “but since you remain in darkness when the candle is withdrawn, you have not light in yourselves.”  (qt’d in Morris p. 319)