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").
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.