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I wanted to know why most translations seem to use "Eternal (or Everlasting) Father" and others use "Father of Eternity" in Isaiah 9:6. I found the following statement in an answer to a different question about the same verse:

אֲבִיעַד (aviad): phrase consisting of the noun אֲבִי in the construct state, meaning "father of," and עַד, meaning "eternity." Altogether, literally meaning "father of eternity," but understood as "eternal father." (cp. Hab. 3:6: , i.e.,
"eternal mountains"

This does help me, but I would still like to know why the phrase is understood as "eternal father" when it literally says "father of eternity".

This is a common function of the Hebrew construct chain without direct equivalent in English. This is normal: syntactic constructions don't map one-to-one to constructions in other languages. Thus, while the most frequent meaning of the construct chain is "X of Y", this does not mean that it is the 'most literal' translation. Indeed, it has functions that cannot be expressed by English "X of Y", because the use patterns of these constructions do not fully overlap.

This particular type of relation is called a genitive of quality (Joüon and Muraoka, §129f):

The other genitives are mainly: [...] the genitive of the quality expressed by an abstract noun (which often makes up for the lack of adjectives): Ex 29.29 בִּגְדֵי הַקֹּדָשׁ the garments of holiness (= the holy garments); Lv 10.17 מְקוּם הַקֹּדָשׁ the holy place (= the sanctuary); Lv 19.36 מֹאזּנֵי צֶדָק scales of justice (just scales); Ex 5.9 דִּבְרֵי שֶׁקֵר deceptive words; 1Kg 20.31 מַלְכֵי חֶסֶד merciful kings; Gn 17.8 אֲחֻזַּת עֹולָם possession in perpetuity; Jdg 11.1 גִּבּוֹר חַיִל valiant warrior; Pr 1.10 לִוְיַת חֵן charming crown.

Waltke and O'Connor call it an attributive genitive (§9.5.3), reserving genitive of quality for cases like שְׁלֹומֶךָ your peace (Isa 48:18; §9.5.1j), a function that is shared by English of. To Joüon and Muraoka's examples they add for the attributive genitive Prov 5:19 אַיֶלֶת אַהָבִים beloved hind, Ps 48:2 הַר־קָדְשׁוֹ his (God's) holy mountain ('mountain of his holiness').

All in all, this is a rather common function of the grammatical construction. To say that "father of eternity" is "more literal" is misleading, because it suggests that semantic functions of grammatical constructions map one-to-one between languages while they do not.

There is definitely no need to ascribe theological meaning to this particular case, or discuss the differences between English father of eternity and eternal father; that is special pleading and does not apply likewise to cases like valiant warrior ('warrior of strength', Jdg 11:1).

A Peculiar Case

The most common interpretation, "Eternal Father", suggests that God exists eternally as a "Father" to all. The literal interpretation, "Father of eternity", means that God is "Father" to all again, but including the notion of eternity and the physical experience/thing of eternity.

The peculiar thing about this problem is that both the phrases Eternal Father and Father of eternity don't mean the exact same thing, but are both true about God. God is both the Eternal Father and the Father of eternity:

“From eternity to eternity I am God. No one can snatch anyone out of my hand.
No one can undo what I have done.” Isaiah 43:13

“I am the Alpha and the Omega,” says the Lord God, who is and was and is to come—the Almighty. Revelation 1:8

Again, God is both the Eternal Father, "the Alpha and Omega", and the Father of eternity, (i.e. "from eternity to eternity I am God").

My Answer

This is going to sound a little informal because I'm not well-versed in Hebrew, but I have a good idea of where the possible answer therein lies.

אֲבִיעַ֖ד (ab-i-ad), a Hebrew word that means literally, "Father of eternity", whereas it is translated as "Eternal Father" in English, has come across as odd to you. To solve this problem, we can look no further than the man who God deemed would be the father to many nations, Abraham. Before his name was changed to Abraham, his name was Abram, which can be translated as, "Exalted Father", or literally as, "Father of elevation". אברם (ab-rum), a Hebrew word that (note) doesn't have the "i" sound/word there. Why is it still translated (literally) as "Father of elevation"?

My guess is that the word "of" doesn't fully encompass the properties that the Hebrew word/syntax implies. To state the more exact literal interpretation of "Father of elevation" would be "Father who is attributed with elevation". So that is all to say that אֲבִיעַ֖ד (ab-i-ad) is more exactly translated literally not as "Father of eternity", but "Father who is attributed with eternity"; that is the most probable answer to why Everlasting/Eternal Father is the more common translation, as it is more close to the original meaning as literal translations often err on the side of fully-fleshed out meaning.

For more information, a link to the website where I found most of my info about Abram is here.

Isa. 9:6 is reffering to Jesus, God's "son," and some of his functions. "Eternal Father" fits best as he is Adam's replacment who could of been an Eternal Human and Spiritual Father to mankind but he forfited that when he sinned in Eden. Jesus can give humans what Adam also gave up:-

NWT John 17:3 "This means everlasting life, their coming to know you, the only true God, and the one whom you sent, Jesus Christ."

NWT 1 Corinthians 15:22 "For just as in Adam all are dying, so also in the Christ all will be made alive."

Added

NWT 1 Corinthians 15:45 "“The first man Adam [In Eden] became a living person.” The last Adam [Jesus] became a life-giving spirit.