What is the basis for the claims that the word "Israel" means "Exiled for their sins"?

According to the Hebrew commentators on the word, "Isra" means "fought" and is derived from the root "sar", and there is a double entendre because it also means lord or prince, "sar". So in this context it seems to mean "lorded over greater powers" - i.e. he won the fight against the angel, i.e. "El" (which can mean any sort of higher power, not necessarily God) described in Genesis 32:25-30, as Evan Harper has aptly explained.

According to Medrashim cited by Rashi, the angel in that case was the "guardian angel" of Esau, whom Jacob was about to encounter after a long separation and considerable animosity. The metaphorical battle with the angel portends Jacob's success in winning over and to some degree healing his old feuds with Esau.

The name Yisroel - Israel is associated with that victory and it is possible that this incident and the bestowing of this special name denotes the birth of Yisroel - the Jewish People, as a nation, rather than just a collection of tribes. The subsequent narrative in B'reishit (Genesis) also supports this. Subsequently Jacob and Esau each went their own respective ways and established their own domains through their descendants: Esau dominating certain areas which are identified perhaps with the Negev and Sinai region or even parts of the Arabian peninsula, while Jacob and his seed eventually ended up in Egypt, following Joseph, who was the first to arrive there and achieved power and prominence according to the biblical account. This account has some support in the archaeological and anthropological record: See pilgrimtours.com/mideast/israel/Info/ExodusNile.pdf - The Historicity of Joseph, and for a more scholarly discussion of confirmation of parts of the biblical account, see "The Jews of Egypt: From Rameses II to Emperor Hadrian" from The Jewish Publication society, a scholarly work.

In short, this title Yisroel - Israel, denotes victory, dominance and the rise of the Jewish People, and is generally used in such manner in the biblical context. The Jews are sometimes referred to as "Israel" and at other times as "Jacob" or "The House of Jacob", and commentators have noted that "Israel" is generally used in laudatory fashion, while "Jacob", denotes a lower sort of status, associated with subservience, defeat, being downtrodden, etc - analogous to Jacob's situation previous to earning the name "Israel" through his victory against the angel and subsequent success in placating his long-hostile brother Esau.

That being the case, I can say with some authority that if Davidovits makes such a claim, it is patently false, based on the original source material. If he has some other source, he must bring it and prove that it trumps what I have stated herein, based on primary, authoritative Hebrew etymology and source material detailed in the classical commentators on the book of B'reishit (Genesis), discussing the derivation of the name Yisroel - Israel.

Sources - Rashi, Targum, Rashbam on B'reishit Kaf Beit.

Isra'el means "he struggles with God" and is the name granted to Jacob after he wrestles with an angel in Genesis 32:

Then Jacob was left alone, and a man wrestled with him until daybreak. When he saw that he had not prevailed against him, he touched the socket of his thigh; so the socket of Jacob’s thigh was dislocated while he wrestled with him. Then he said, “Let me go, for the dawn is breaking.” But he said, “I will not let you go unless you bless me.” So he said to him, “What is your name?” And he said, “Jacob.” He said, “Your name shall no longer be Jacob, but ^[n]Israel; for you have striven with God and with men and have prevailed.”

[n]: I.e. he who strives with God; or God strives

Jacob is said to be the ancestor of all the tribes of Israel, to whom his name attached.