The "Tetragrammaton" name

A theophany is a manifestation of God in the Bible that is tangible to the human senses. In its most restrictive sense, it is a visible appearance of God in the Old Testament period, often, but not always, in human form. Frequently, the term “glory of the Lord” reflects a theophany, as in Exodus 24:16-18; the “pillar of cloud” has a similar function in Exodus 33:9. A frequent introduction for theophanies may be seen in the words “the Lord came down,” as in Genesis 11:5; Exodus 34:5; Numbers 11:25; and 12:5.

However, when the pre-incarnate Christ has appeared, that is called a Christophany. These appearances are usually identified by the expression “the angel of the Lord” and are found in Genesis 16:7-14; Genesis 22:11-18; Judges 5:23; 2 Kings 19:35; and other passages. While there are no indisputable Christophanies in the Old Testament, every theophany wherein God takes on human form foreshadows the incarnation, where God took the form of a man to live among us as Emmanuel, “God with us” (Matthew 1:23).

Since any pre-incarnate appearance of Christ in the Old Testament is called a Christophany and not a Theophany, then I really don’t know how this links to the Tetragrammaton (YHWH) in the Old Testament. Perhaps you could provide some additional information from your research?

EDIT: Thank you for clarifying your question. With regard to the theophany in Genesis 12:7, my NIV Study Bible (2000 edition) makes this comment:

The LORD appeared: The Lord frequently appeared visibly to Abram and to others, but not in all his glory (see Exodus 33:18-20; John 1:18).

“The LORD came down” is the clue that indicates it was a Theophany. The NIV Study Bible notes on the incident in Numbers 12:3 when the Lord came down in a pillar of cloud to speak with Moses, Aaron and Miriam make this comment:

Came down – Often used of divine manifestations. In Numbers 11:5 he came down in grace; here and in Genesis 11:5 he came down in judgment. In a sense every theophany (appearance of God) is a picture and promise of the grand theophany, the incarnation of Jesus, both in grace and judgment.

There is another incident (Joshua 5:13) where Joshua “saw a man standing in front of him.” The NIV Study Bible notes say this:

The experience is taken by many to be an encounter with God in human form (theophany) or with Christ (a Christophany). But angels also were sent on missions of this kind (Judges 6:11; 13:3), and some were identified as captains over the heavenly armies (Daniel 10:5, 20; 12:1).

Yes, because Jesus is part of the One Being of God, there is a sense in which we can say a theophany is also a Christophany – but not with absolute certainty. Opinion is divided. The incident in Genesis 12:7 is positively identified as a theophany, but not all appearances are so clear cut. Here are some Old Testament examples:

Genesis 18:1-33 – One day, Abraham had some visitors: two angels and God Himself. He invited them to come to his home, and he and Sarah entertained them. Many commentators believe this could also be a Christophany, a pre-incarnate appearance of Christ. Genesis 32:22-30 – Jacob wrestled with what appeared to be a man, but was actually God (vv. 28-30). This may also have been a Christophany. Exodus 3:2 - 4:17 – God appeared to Moses in the form of a burning bush, telling him exactly what He wanted him to do. Exodus 24:9-11 – God appeared to Moses with Aaron and his sons and the seventy elders. Deuteronomy 31:14-15 – God appeared to Moses and Joshua in the transfer of leadership to Joshua. Job 38–42 – God answered Job out of the tempest and spoke at great length in answer to Job’s questions.

When the Bible says “the angel of the Lord” appeared, it means a Christophany. The important thing to remember is that after the incarnation (Matthew 1:23) the appearances of “the angel of the Lord” cease after the incarnation of Christ. Angels are mentioned numerous times in the New Testament, but “the angel of the Lord” is never mentioned in the New Testament after the birth of Christ.
Source: https://www.gotquestions.org/angel-of-the-Lord.html