Two patriarchs have their named changed. Abram becomes Abraham and Jacob becomes Israel:
Neither shall thy name any more be called Abram, but thy name shall be Abraham; for a father of many nations have I made thee. (Genesis 17:5 KJV)
And he said, Thy name shall be called no more Jacob, but Israel: for as a prince hast thou power with God and with men, and hast prevailed. (Genesis 32:28 KJV)
And God said unto him, Thy name is Jacob: thy name shall not be called any more Jacob, but Israel shall be thy name: and he called his name Israel. (Genesis 35:10 KJV)
After his name is changed Abram is always called Abraham. Despite two statements that Jacob shall be called Israel, he continues to be called Jacob and sometimes by both names in the same context:
And God spake unto Israel in the visions of the night, and said, Jacob, Jacob. And he said, Here am I. (Genesis 46:2 KJV)
Sometimes the names of the patriarchs are used to identify "their" God:
And he said, “I am the God of your father, the God of Abraham, the God of Isaac, and the God of Jacob...” (Exodus 3:6 KJV)
Go and gather the elders of Israel together and say to them, ‘The Lord, the God of your fathers, the God of Abraham, of Isaac, and of Jacob, has appeared to me... (Exodus 3:16 KJV)
Since the LORD is speaking to Moses after the death of Jacob/Israel, one would expect Him to acknowledge Jacob by his changed name. Yet He calls the man Jacob. In doing so He establishes a continuity with the patriarchs which at the same time is form of self-identification. It is this phrase which is carried on almost exclusively through the Old and New Testaments.
However, there are 3 occasions when Israel replaces Jacob:
And it came to pass at the time of the offering of the evening sacrifice, that Elijah the prophet came near, and said, Lord God of Abraham, Isaac, and of Israel, let it be known this day that thou art God in Israel, and that I am thy servant, and that I have done all these things at thy word. (1 Kings 18:36 KJV)
O Lord God of Abraham, Isaac, and of Israel, our fathers, keep this for ever in the imagination of the thoughts of the heart of thy people, and prepare their heart unto thee (1 Chronicles 29:18 KJV)
So the posts went with the letters from the king and his princes throughout all Israel and Judah, and according to the commandment of the king, saying, Ye children of Israel, turn again unto the LORD God of Abraham, Isaac, and Israel, and he will return to the remnant of you, that are escaped out of the hand of the kings of Assyria. (2 Chronicles 30:6 KJV)
Each of these 3 come at significant moments. In particular, the first time the phrase is used in by Elijah during the testing of the prophets of Ba'al.
What is the significance to saying the God of Abraham, Isaac, and Israel rather than the God of Abraham, Issac, and Jacob?