Every individual has a mission in this world, and one’s name expresses his unique mission. After our patriarch Jacob fulfilled the mission of his name Jacob, he was called “Israel,” and was given an additional mission to fulfill in this world.

One’s name alludes to the mission that he must fulfill in this world:

One’s name is that matter that he was created to rectify; he was named for this. Therefore, evil people [who do not rectify that which they are supposed to rectify during their lifetimes] forget their names in the grave. But Jacob, not only did he rectify that for which he was named, but he merited being called by the name Israel as well [see Genesis 32:29].

A person is engaged in a mission all his days, and he is named for it:

All his days, a person is engaged in his mission, until he fulfills the mission and departs to the upper world. Jacob, who had already rectified that for which he was named, was given another name. (Rabbi Yehuda Aryeh Leib Alter, Sefat Emet, Vayishlaĥ 5657)

One should pray that all those who are called by his name will have these good attributes, and not one of the evil attributes. (Rabbi Yehuda HeĤasid, Sefer Ĥasidim)

Further reading: For the halakhot that relate to naming, see A Concise Guide to Halakha, pp. 7, 15 (for boys), and p. 30 (for girls).