The word ישראל Yishrael (Israel) has several meanings,
Type of meaning
The Prophet Israel
Jacob had wrestled with an angel, and refused to let the angel go even after the angel struck Jacob and injured him to try and defeat him. The angel asked to let go, and Jacob agreed, but only if he receives a blessing, and the angel blesses him by changing Jacob’s name to Israel. There are several interpretations to how being given this name was a blessing.
Jacob was not unscathed. This happened at night. That is a tough setting — literally, a dark moment.
There is a symmetry in Torah — there are comparable events in each era, cycles of behavior that repeat in each generation, for example, to deal with the patriarchs (e.g. Jacob) and the nation (e.g. Judah). The way the community could argue with G-d and gain concessions, so did the individual Jacob engage in a physical combat with an angel.
Also, people’s closest relationships often involve struggle. This allows for growth and closeness. As such, that relationship must be characterized in Jacob’s relationship with G-d — in other words, as the Torah makes clear, between Israel’s relationship with G-d.
It was not necessarily a violent wrestling match. It may have also been playful. However, it reached an intensity that resulted in harm to Jacob, the human in the match.
Sons of Israel
People of Israel
This is a national term for the Jewish people.
Land of Israel
State of Israel
A modern republic founded in 1948 as a homeland for Jewish people, to represent Jewish people on international matters, and to guarantee freedom and equality on domestic matters for all its Jewish and non-Jewish citizens alike.
Some Christian and other groups appropriate the Jewish use of the term to suggest they are authorities as opposed to or even over authentic Jewish groups. Examples include the so-called Black Hebrew Israelites, Messianic Christians, and so forth.