In other words, is Exodus 3:14 suggesting that the Tetragrammaton is a form of the verb "to be"? Can someone explain the linguistic path from the Tetragrammaton to the verb, "to be"?
The noun היה/הוה means "to be" (see this), if you add the prefix alef אהיה it means "i will be", if you add instead a yud יהיה it means "he will be". This is the the meaning of the Tetragrammaton according to the Rashbam exodus 3:15 (written in codes א"ת ב"ש).
It seems that most interpreters took it as form of "to be", but they disagree as to what this expression is supposed to convey.
Rashi (which almost always reflects the views of the Rabbis) interprets it meaning that i will always be with you Israel; i will take you out of egypt and even of other exiles. According to this explanation the Tetragrammaton is understood in the context of the exodus only, and the name is supposed to convey something about their condition in egypt and the exodus that will follow.
However, Rashbam, Chizkuni and Sforno understood this more broadly, they interpret it meaning i will be or exist forever. The name reveals more about the nature of god himself, and the interpretation is more philosophical, and less focused on the condition in egypt, though it conveys something about their exodus as well.
So you see that this linguistic relationship has not been lost to us, on the contrary it has been understood so by our Rabbis and later Jewish commentators for the past 2000 years.
Yes, very well-known. During prayers, upon encountering the Tetragrammaton, most Jewish prayer books note that the proper concentration is -- "was, is, and will be." -- I.e., God is above time.
YHVH This is the Tetragrammaton which may not be pronounced under any circumstances (cf. Sanhedrin 90a; Philo, De Vida Moses 3:519, 529). If this section is read out loud, this name should be read as 'Lord' (cf. Septuagint). This name denotes God's utter transcendence (Kuzari 2:2; Moreh Nevukhim 1:61). This name also denotes the creative power that constantly sustains the universe. God is telling Moses that not only is the initial purpose of creation now being fulfilled, but also the process that will insure its continual existence.
Sorry about the long quote. I didn't want to degrade N. M. Sarna's excellent coverage of this topic. Please credit this answer to N. M. Sarna.
Ehyeh-Asher-Ehyeh This phrase has variously been translated, “I Am That I Am,” “I Am Who I Am,” and “I Will Be What I Will Be.” It clearly evokes YHVH, the specific proper name of Israel’s God, known in English as the Tetragrammaton, that is, “the four consonants.” The phrase also indicates that the earliest recorded understanding of the divine name was as a verb derived from the stem h-v-h, taken as an earlier form of h-y-h, “to be.” Either it expresses the quality of absolute Being, the eternal, unchanging, dynamic presence, or it means, “He causes to be.” YHVH is the third person masculine singular; ehyeh is the corresponding first person singular. This latter is used here because name-giving in the ancient world implied the wielding of power over the one named; hence, the divine name can only proceed from God Himself.
In the course of the Second Temple period the Tetragrammaton came to be regarded as charged with metaphysical potency and therefore ceased to be pronounced. It was replaced in speech by ʾadonai, “Lord,” rendered into Greek Kyrios. Often the vowels of ʾadonai would later accompany YHVH in written texts. This gave rise to the mistaken form Jehovah. The original pronunciation was eventually lost; modern attempts at recovery are conjectural.
God’s response to Moses’ query cannot be the disclosure of a hitherto unknown name, for that would be unintelligible to the people and would not resolve Moses’ dilemma. However, taken together with the statement in 6:3, the implication is that the name YHVH only came into prominence as the characteristic personal name of the God of Israel in the time of Moses. This tradition accords with the facts that the various divine names found in Genesis are no longer used, except occasionally in poetic texts; that of all the personal names listed hitherto, none is constructed of the prefixed yeho-/yo- or the suffixed -yahu/-yah contractions of YHVH; that the first name of this type is yokheved (Jochebed), that of Moses’ mother. Ibn Ezra points out that Moses, in his direct speech. invariably uses the name YHVH, not ʾelohim, “God.” Without doubt, the revelation of the divine name YHVH to Moses registers a new stage in the history of Israelite monotheism. -- Sarna, N. M. (1991). Exodus (pp. 17–18). Philadelphia: Jewish Publication Society.
It is not necessarily linked linguistically, the teaching of Rabbi’s isn’t that from God. Moses question in Ex. 3:13 is,”What shall I say to them?”, prefixed by a hypothetical question of the Israelites asking him, “What is His name?”. In Ex. 4:1, Moses proposed response of the Israelites, “YHWH has not appeared unto you”, answers what part of all he was told to say, is The Hallowed Name. The eternal GOD chose His name long before. Also note that in the middle of Moses question in 3:13, there is an end forward acrostic spelling The Hallowed Name. It is not a coincidence that there is a perfect set of such acrostics in this small portion of text. Ex. 4:3, 4:14 and 4:16 complete a set similar to that in Esther, but that’s another matter. In Ex. 3:12 Moses is assured by GOD, “Certainly I will be...”, the same as he is told to assure the Israelites in 4:14 (ehyeh).