In the tanakh Avraham, Yitzchak, and Yaakov are the three patriarchs of the Jewish people, beginning with a covenant with Avraham (Gen 12, 15, 17) and culminating in God revealing himself and giving his torah (Ex 20) to the family of Israel (i.e. Yaakov, after God's messenger renamed him (Gen 32)). God speaks directly to these three patriarchs.
God introduces himself as "God of Avraham [, God of Yitzchak, God of Yaakov]" several places in torah, including:
Gen 26:24, to Yitzchak: "I am the God of Avraham your father; fear not..."
Gen 28:23, to Yaakov in a dream, as he fled his brother Eisav: "I am the Lord God of Avraham your father, and the God of Yitzchak..."
Ex 3:6, to Moshe at the burning bush: "I am the God of your father, the God of Avraham, the God of Yitzchak, and the God of Yaakov..."
Ex 3:15-16, similarly (God telling Moshe what to tell the people)
Ex 4:5, instructing Moshe to throw down his staff so it might become a snake: "...that they may believe that the Lord God of their fathers, the God of Avraham, the God of Yitzchak, and the God of Yaakov, has appeared..."
Moshe appeals to this lineage when pleading for the people after the golden calf in Ex 32:13: "Remember Avraham, Yitzchak, and Yisrael, your servants, to whom you swore...".
God also refers to his promises to this trio when condemning the people to die in the wilderness in Num 32:11: "Surely none of the men... shall see the land which I swore to Avraham, to Yitzchak, and to Yaakov...".
Moshe makes many references to the patriarchs as a group when addressing the people in Deuteronomy.
So, putting all that together: God had a special relationship and covenant with the patriarchs, and through them the whole house of Israel, and this history is recalled at key moments. Sure, God is also the God of Moshe and Aharon and everybody else (even me!), but Avraham, Yitzchak, and Yaakov have a special place.
Presumably Jesus, being a Jew, intended to remind his Jewish followers of God's role using the same words that were used in the scriptures with which they were all assumed to be familiar.