Yes, God has a personal name. When Moses was going up to the mount to see the burning bush, he encountered God. Let's take a look at what the Bible says happened.
"Then Moses said to God, 'Behold, I am going to the sons of Israel, and I shall say to them, 'The God of your fathers has sent me to you.' Now they may say to me, 'What is His name?' What shall I say to them?' 14 And God said to Moses, 'I AM WHO I AM'; and He said, 'Thus you shall say to the sons of Israel, 'I AM has sent me to you,'' (Exodus 3:13-14).
As you can see, God declares His own name to be 'I AM.' However, some people say that the name of God is Jehovah or Yahweh. The reason this is said is that in Hebrew (the book of Exodus was written in Hebrew), the word for 'I AM' is אֶהְיֶה which is YHWH. These four letters are known as the Tetragrammaton. This is why a lot of English translations will say that the name of God is Jehovah or Yahweh. It is an English styled version of those four letters.
Nevertheless, God's name is 'I AM.' It is His personal name.
The name of God reflects His character, His existence, His Majesty, etc. Those of us who speak English don't really understand the sense in which the names used in Scripture carry meaning. One of the best ways we can understand this concept is with American Indians. They would have names like "Running Bear, " "Flying Hawk," etc. The names had a kind of signification that was supposed to represent, in one form or another, the character of the person. The same concept is applicable in Scripture where the names of people have meaning. For example, Noah means "hope, peace." Methuselah means "when he dies it will come." The Scripture is full of such types of names that have meaning.
Other "Names" of God
The Bible uses other titles and names for God, not in the proper name sense of "I Am." Some of the other names and titles of God are:
El Shaddai--"God Almighty" This title first occurs in Genesis 17:1 where God is renewing His covenant with Abraham.
El Elyon "God is Exalted" or "God most High" This title first occurs in Genesis 14:18-21 in the encounter between Abraham and Melchizedek. See also Deuteronomy 26:19; 2 Chronicles 7:21.
El Olam "The Everlasting God" This title first occurs in Genesis 21:33.
El Gibbor "God is Mighty"
The Ancient of Days is found in Daniel 7:9, 13, 22
The Living God is mentioned in 1 Samuel 17:26; Isaiah 37:17; Psalm 42:2; Jeremiah 10:10
The Lord God is used in Revelation 1:8; 22:5
The Lord God Almighty is used in Psalm 89:8
The Lord Most High is used in Psalm 7:17
Etc : https://carm.org/does-god-have-a-name