I and other people, including Catholics, hope to revive this identification, not only to better connect with the Church Fathers, but also to better savor the Holy Scriptures, since if the Angel of the Lord is Jesus, then that brings out the Trinity within the earliest parts of the Old Testament!
The first and most important thing to keep in mind when approaching the "Angel of the Lord" issue is that both the Hebrew and the Greek term for "angel" is actually a generic word for "messenger". Nothing about either term actually specifically refers to the spirit creature with wings that we normally think about when the term "angel" is used. In fact, when the original languages were used, the texts simply read the "Messenger of the Lord". A Messenger could be anything from an angel to a prophet to a teacher to an ambassador. Even names like of the prophet "Malachi" are simply the Hebrew word for "Messenger".
- Justin Martyr: "And that Christ being Lord, and God the Son of God, and appearing formerly in power as Man, and Angel, and in the glory of fire as at the bush, so also was manifested at the judgment executed on Sodom, has been demonstrated fully by what has been said." (Dialogue with Trypho, ch128)
- Irenaeus: "And again, when the Son speaks to Moses, He says,
I have come down to deliver this people.
Exodus 3:8 For it is He who descended and ascended for the salvation of men." (AH 3.6.2) - Tertullian: "Thus was He ever learning even as God to converse with men upon earth, being no other than the Word which was to be made flesh. But He was thus learning (or rehearsing), in order to level for us the way of faith, that we might the more readily believe that the Son of God had come down into the world, if we knew that in times past also something similar had been done." (Against Praxeas, ch16)
- Clement of Alexandria: "Now that the Word was at once Jacob's trainer and the Instructor of humanity [appears from this]--"He asked," it is said, "His name, and said to him, Tell me what is Try name." And he said, "Why is it that thou askest My name?" For He reserved the new name for the new people--the babe; and was as yet unnamed, the Lord God not having yet become man. Yet Jacob called the name of the place, "Face of God." "For I have seen," he says, "God face to face; and my life is preserved." The face of God is the Word by whom God is manifested and made known. Then also was he named Israel, because he saw God the Lord. It was God, the Word, the Instructor, who said to him again afterwards, "Fear not to go down into Egypt."" (The Instructor, 1:7)
- Eusebius: "Remember how Moses calls the Being, Who appeared to the patriarchs, and often delivered to them the oracles afterwards written down in Scripture, sometimes God and Lord, and sometimes the Angel of the Lord. He clearly implies that this was not the Omnipotent God, but a secondary Being . . . the Being Who was seen must have been the Word of God, Whom we call Lord as we do the Father. But it is needless for me to labour the point, since it is possible to find instances in Holy Scripture. These I will collect at leisure in connection with my present work to prove that He Who was seen by the patriarchal saints was none other than the Word of God." (Proof of the Gospel, 1.5)
- Ambrose: "It was not the Father Who spoke to Moses in the bush or in the desert, but the Son. It was of this Moses that Stephen said,
This is He Who was in the church, in the wilderness, with the Angel.
Acts 7:38 " (Exposition of the Faith, 1:13:83) - Hilary: "For God appeared from the bush as the Angel of God, and the prayer for Joseph is that he may receive such blessings as He shall please. He is none the less God because He is the Angel of God; and none the less the Angel of God because He is God. A clear indication is given of the Divine Persons; the line is definitely drawn between the Unbegotten and the Begotten." (On the Trinity, 4:33)
- Theodoret: "The whole passage (Exodus 3) shows that it was God who appeared to Moses. But Moses called Him an “angel” in order to let us know that it was not God the Father whom he saw but the Only-begotten Son, the Angel of Great Counsel." (quoted in Catholic Encyclopedia "Angels")
- Genesis 16:7-14. The Angel of the Lord appears to Hagar, but the Angel speaks as if He is God Himself, saying things such as "I will multiply your offspring." At the end of the message Hagar replies, "Truly I have seen [God] who looks after me."
- Genesis 21:7-19. Again the Angel of the Lord appears to Hagar and speaks like God Himself: "I will make [Ishmael] into a great nation."
- Genesis 22:11-18. The Angel of the Lord appears to Abraham and says: "I know that you fear God, seeing you have not withheld your only son from me" and "By myself I have sworn, declares the Lord, because you have done this I will surely bless you, because you have obeyed my voice".
- Genesis 31:11-13. The "Angel of God" says to Jacob: "I am the God of Bethel, where you anointed a pillar and made a vow to me."
- Exodus 3:2-6. The Angel of the Lord appears to Moses in a burning bush, where "God called to him out of the bush" and said to Moses: "I am the God of your father, the God of Abraham, the God of Isaac, and the God of Jacob." The passage even says: "And Moses hid his face, for he was afraid to look at God."
- Judges 2:1-5. The Angel of the Lord appears and says: "I brought you up from Egypt and brought you into the land that I swore to give to your fathers. I will never break my covenant with you".
Also worth considering are: Genesis 18:1-22 where 'angel messengers' appear as men to Abraham; similarly Gen 19:1; in Gen 32:24-30 there's the famous encounter with Jacob who "wrestles" with a "man" (Hosea 12:3-4 says it was an "angel") and Jacob was named Israel (meaning "wrestles with God") and said he saw the face of God; in Gen 48:15-16 mention is made of 'angel' alongside God's work of salvation; in Exodus 14:19-22 both the 'Angel of God' and the Lord are mentioned saving the Israelites from the Egyptians; in Joshua 5:13-15 a "man" appears to Joshua and claims to be captain of God's army, and tells Joshua that this is holy ground; in Judges 6:11-24 the Angel of the Lord and has a talk with Gideon, which ends with Gideon afraid of seeing God and builds an altar there in commemoration of God visiting him; in Judges 13:2-23 the Angel of the Lord appears and has a long discussion with a barren couple, the dialog distinguishes between God and the Angel but the Angel speaks like God Himself; in Zech 3:6-10 it recalls when the Angel of the Lord appeared and talked to Joshua, but the dialog is as if God Himself is speaking.
We also know that often times God simply speaks directly to people (e.g. Gen 4:6; 6:13; 12:1; 17:1; 20:3), without the need for an angel to act as an ambassador between God and men. Also, we know that sometimes angels (plural or singular) show up without any message. So God making use of the "Angel of the Lord" to speak in God's Person is highly mysterious, and it seems the best answer for the mature Christian is to accept that it is the Son of God.
I'm not sure what else to say, other than this has been a fascinating research for me. On one hand I am eager to make this identification, because the Church Fathers do and the mysterious language of the OT references; yet on the other hand, I'm not overwhelmingly convinced given that the New Testament doesn't explicitly make the identification, which you'd think would be so obvious for apologetic and preaching purposes.