According to multiple sources, this hymn is based on The Yigdal of Daniel ben Judah, a Jewish judge in Rome around the year 1400.  It was paraphrased by Thomas Olivers in the eighteenth century.

From CyberHymnal:

One night in Lon­don, [Ol­iv­ers] was at­tract­ed to a ser­vice in a Jew­ish syn­a­gogue, where he heard a great sing­er, Le­o­ni, sing an an­cient He­brew mel­o­dy in the sol­emn, plaint­ive mode and he be­came im­pressed with a de­sire to write a hymn to that tune. The re­sult was our hymn, “The God of Abra­ham Praise,” which in a sense is a par­a­phrase of the an­cient He­brew Yig­dal, or dox­ol­o­gy, though Olivers gave to it a dis­tinct­ly Christ­ian fla­vor.

The sto­ry is told of a young Jew­ess who had been bap­tized in­to the Christ­ian faith, and in con­se­quence was aban­doned by her fam­i­ly. She fled to the home of the min­is­ter, poured out her heart to him, and as if to show that, af­ter all, her joy in her new-found Sav­iour was great­er than all her loss of home and fam­i­ly, she sang, “The God of Abra­ham Praise.”

When we compiled our hymnal, we began with the four verses most commonly used in the hymnals we had at the time.  The hymn was originally Jewish, and I felt it needed to speak a little more directly of Christ than the four stanzas we had.  The third verse, according to Cyberhymnal, was thus:

The God of Abraham praise, whose all sufficient grace
Shall guide me all my happy days, in all my ways.
He calls a worm His friend, He calls Himself my God!
And He shall save me to the end, thro’ Jesus’ blood.

That was better, but not quite what I wanted, so I constructed a new third verse.  I began with Christ’s great claim in John 8:58, “Before Abraham was, I am!”  Nothing could better identify our Saviour as being the God of Abraham, which is the theme of the hymn.  I next drew on John 1:1 (“and the Word was God”), using the words, “the God of power and might.”

Later in John 1, at verse 14, we see that He became flesh, for the purpose (v 29) of being the Lamb of God.  Furthermore, Abraham had said God would provide Himself a Lamb (Genesis 22:8) for a sacrifice, so I thought, in keeping with the “God of Abraham” theme, it would be good to mention the Lamb for Whom Abraham had looked.  “Became the sacrificial Lamb….”

Now, since John 1 and John 8 were working together, the words “Who gave us light” pushed themselves to the forefront, from John 1:4-5 (“the Light shineth in darkness”) and John 8:12 (“I am the light of the world”).

The last half of our version is quite similar to the original third verse.  I changed “worm” to “slave” to parallel John 15:15:  “Henceforth I call you not servants; for the servant knoweth not what his lord doeth: but I have called you friends; for all things that I have heard of my Father I have made known unto you.”  I tend to get emotional when I sing those words — “He calls a slave His friend, He calls Himself my God!”  Truly, His grace is amazing, for we are not even worthy to be His slaves, but He loves us.

The net result may not have been a poetic masterpiece :), but it packed the content of multiple Scriptures into a third verse which, I believe, fits nicely with the theme of the hymn, is doctrinally sound, and makes the hymn much more Christological than it would have been with only the other four verses we were using.  So this is what we sing in our church:

The God of Abraham praise, Who reigns enthroned above;
Ancient of everlasting days, and God of Love;
Jehovah, great I AM! by earth and Heav’n confessed;
I bow and bless the sacred Name, forever blessed.

 The God of Abraham praise, At whose supreme command
From earth I rise—and seek the joys at His right hand;
I all on earth forsake, its wisdom, fame, and power;
And Him my only Portion make, my Shield and Tower.

‘Before Abraham, I am!’  The God of power and might,
Became the Sacrificial Lamb, Who gave us light.
He calls a slave His friend, He calls Himself my God!
I shall be saved unto the end, thro’ Jesus’ blood.

 He by Himself has sworn; I on His oath depend,
I shall, on eagle wings upborne, to Heav’n ascend.
I shall behold His face; I shall His power adore,
And sing the wonders of His grace forevermore.

The whole triumphant host Give thanks to God on high;
“Hail, Father, Son, and Holy Ghost,” they ever cry.
Hail, Abraham’s God, and mine! (I join the heav’nly lays,)
All might and majesty are Thine, and endless praise.

The music can be heard by clicking on the video: