Calling God “Abba, Father” is different from giving God a familiar name. Calling God “Abba” is entering into the same intimate, fearless, trusting, and empowering relationship with God that Jesus had. That relationship is called Spirit, and that Spirit is given to us by Jesus and enables us to cry out with him, “Abba, Father.”
Calling God “Abba, Father” (see Roman 8:15; Galatians 4:6) is a cry of the heart, a prayer welling up from our innermost beings. It has nothing do with naming God but everything to do with claiming God as the source of who we are. This claim does not come from any sudden insight or acquired conviction; it is the claim that the Spirit of Jesus makes in communion with our spirits. It is the claim of love.