After Moses grasps the serpent by the tail and it becomes a rod, the Holy One explains the purpose of the miracle. Targum Pseudo-Jonathan puts it plainly: In order that they may believe that the Lord God of their fathers, the God of Abraham, the God of Izhak, and the God of Jakob, hath revealed Himself to thee.
Notice what the sign is for. Not to prove God's power — the slaves in Egypt have been witnesses to Nile floods and royal architecture their entire lives. Power they understand. The sign proves something else: that the God speaking to Moses is the same God who spoke to Abraham, to Isaac, to Jakob.
A Chain of Three Names
The Targum lingers on the three patriarchal names because continuity is the real miracle. Four hundred years have passed (Genesis 15:13). An entire generation of enslaved Hebrews has never known a prophet. The God of their great-great-grandparents has been silent. The serpent-rod, the leprous hand, the water-turned-blood — these are not displays of might. They are proofs of identity.
The takeaway: Moses' mission begins not with a new revelation but with a reintroduction. Before the plagues, before the splitting of the sea, before Sinai, the first task is to reconnect a traumatized generation to a God they had almost stopped believing was still listening.