What is the name of God the Father?

The scriptures do not give us a personal name that belongs distinctly to God the Father. In fact, the only person in the Trinity who is given a distinct personal name is God the Son, whose name is Jesus. Even this name is given to Him specifically because of the incarnation. God the Son was not called Jesus until He came in the flesh as the promised Messiah. Neither the Father nor the Holy Spirit are given distinct, personal names.

Some have argued that the Divine Name, YHWH (Yahweh, or Jehovah), is the name of the Father. This name, however, is not exclusively the Father's name. It is the name of the Triune God as a whole. The Father is called YHWH in places like Psalm 110:1

"The LORD [YHWH] says to my Lord [Adoni]: 'Sit at My right hand Until I make Your enemies a footstool for Your feet.'"

Hebrews 1:13 explains this verse as the Father (the LORD, or YHWH) speaking to the Son (the Lord, Adoni, or master). So the author of Hebrews plainly identifies the Father as YHWH. Yet, when Hebrews 1:10-12 quotes Psalm 102:25-27, it says the Psalm is talking about Jesus. The Psalm is clearly talking about YHWH. Jesus, therefore, is also YHWH. Hebrews 10:15-16 quotes Jeremiah 31:33 as being a direct quote from the Holy Spirit. It is, in fact, a direct quote from YHWH. So the Holy Spirit is also YHWH. Just looking at the book of Hebrews alone, we can see that YHWH is the name of the Father, the Son, and the Spirit. Thus, YHWH is the name of the Triune God.

The Father does have titles used only of Him, like "The Ancient of Days" in Daniel 7. Often in the New Testament, the Father is simply called "God" while Jesus is called "the Lord," though this is a general principle rather than a strict rule. The Father is certainly called "the Lord" on occasion (such as Mark 13:20), and Jesus is called "God" in several passages (such as Titus 2:13 and 2 Peter 1:1). Ultimately one must pay careful attention to the context.

In the end, there is one God who exists in three persons, and those three persons share the one divine name of YHWH. That is the name of the Father, but it is also the name of the Son and the Holy Spirit. Three persons, but one God with one Holy Name. The Father, therefore, is not given a separate personal name.