The Haggadah quotes from Mekhilta Ex. 14:31.:

In Egypt it says of them, "The magicians said to Pharaoh `This is the finger of G-d.' (Shemot 8:15) At the sea it says, "Israel saw the great hand that the L-rd laid against Egypt; and the people feared the L-rd, and they believed in the L-rd and in His servant Moses." (Shemot 14:31) Now, how often were they smitten by `the finger'? Ten plagues. Thus you must conclude that in Egypt they were smitten by ten plagues, at the sea they were smitten by [the hand with] fifty plagues.



Aryeh Kaplan offers the possibility that the "finger of G-d" is related to the ancient Egyptian word etzba or tzaba, denoting both finger and retribution.

The Avudraham takes issue with the finger/hand idea since the finger is only mentioned in one place and G-d’s hand is mentioned in another. However, the hand is mentioned by the 5th plague, the culmination of 5 fingers. Also, it says that Hashem took us out of Egypt with a mighty hand and outstretched arm, 2 hands. (Emanuel)

R. Yosi quotes va-yaminu etc. The word va-yaminu occurs one other place in Tanakh, Yonah 3:5. This is when the people of Nineveh respond to Yonah's prophecy. This corresponds to the midrash (Midrash Socher Tov) that says that Pharaoh survived the Red Sea and became king of Nineveh. Like the Israelites, after the miracle, he believed in God and repented when confronted by Yonah. (Baal ha-Turim)

The Ibn Ezra turns this idea on its head. He feels that when the magicians refer to the finger of God, they mean the god of nature whom they did not identify with the God of the Israelites. That puts their statement in an entirely different light.