Of all of the names of God I believe there is none as meaningful and at the same time meaningless as the "I AM". This is because most of His names are defined in terms which are at least partially related to mankind. This relationship gives us a handle or anchor upon which we can base our understanding of the rest of the name. I AM or the fuller form I AM THAT I AM seems to be a definition of God outside of His creation and thus to fully understand it would be to understand something completely outside of our experience. Whilst that task is impossible this side of eternity I shall attempt in this essay to touch upon some of the ideas that this may encompass in the hope that God will be a little clearer and dearer in our sight.
There is even some disagreement as to the correct semantic translation of this name. Whilst I AM THAT I AM is the standard rendering of the Hebrew in Ex 3:14 the Vulgate went with "I am who am". The Septuagint preferred "I am he who exists". In an attempt to capture the meaning some went as far as "The Eternal who passes not away" or even "He who spoke, and the world was; who spake, and all things existed.". Clarke suggests the literal rendering is actually "I will be what I will be" and that some have taken from that that the simply means that God will continue to treat people the way He has thus far.
However, I believe that to focus purely upon the eternal side of I AM and especially to focus upon it in the context of mankind is a mistake. The Jews were already extremely familiar with the term Jehovah having heard it 141 times in Genesis alone. Jehovah is generally taken to mean: "who was, and is and is to come". Some take this similarity of definition to assert that I AM THAT I AM is synonymous to JEHOVAH.
I do not accept that the two terms are fully synonymous. We know that all scripture is inspired and profitable - if God gave us a different for Himself then there is a reason for it. I suggest that Ex 3:15 is suggesting that Jehovah is a reasonable rendering of I AM in relation to Israel. In verse 14 it was the "I AM" that has sent Him; then in verse 15 we get an extra message that the Jehovah God of your fathers is the one sending him. Interestingly the Family Bible Notes on Ex 3:14 suggests a similar division but from the reverse viewpoint in referring to I AM THAT I AM as describing the nature of Jehovah.
In terms of answering the question given; my belief makes this very difficult. It was the term Jehovah that was fully developed throughout the Old Testament and was the root of many of the names of God such as Adon Jehovah, Jehovah Sabaoth and Jehovah Jehovah El.
Barnes asserts that Ps 90:2, Isa 44:6 & 47:8 are oblique references to the title. I think may well be true but it still leaves the I AM as a minor-term in the development of Jewish thought. I believe that I AM THAT I AM was one of those puzzling terms that was largely left alone; possibly this was part of the Jewish downfall.
John Wesley in his notes on Ex 3 also sees I AM as slightly different from Jehovah. He gives I AM 3 attributes; the second is the same as Jehovah and the third is that God will not change. The first attribute is the one I believe is closest to the truth. He suggests it means self-existent, self-sufficient and thus all sufficient. In other words you cannot define the I AM in relation to anything else; He is it. The self defining, self referencing center and total of everything.
I consider it no accident that when the Lord makes one of His more blatant claims to deity in John 8:58 that it is the "I AM" that He refers back to. Of course there are other "I AM""s in John that are the feature of many a Sunday School story but I believe that 8:58 is the one that is supposed to calibrate the claim. In John 8 the Jews are trying to apply human logic and reasoning and some Old Testament theology in order to quantify or pigeon hole the Lord. So He simply takes a step back and declares: "I AM". He exists outside and beyond any thought, reason or inkling of comprehension that they might have.
The response to this declaration in John 8:59 is actually rather funny. Here is the I AM; so they picked up stones to throw at Him. The Lord switches from possibly His grandest to possibly one of His humblest. Jesus hid himself. This is one of those essays that feels woefully inadequate. I just hope that some of the awe of the "I AM" has settled upon you. Almost by definition there is no lesson that can be derived; this really is something outside of us. So instead I will end with the words of the psalmist; if this has induced a little awe inside us - then there is a required response.
Ps 4:4 Stand in awe, and sin not: commune with your own heart upon your bed, and be still. Selah