The short answer is that the word Judaism is a mistake in the same sense that the word "Indian" to describe Native Americans is a mistake. But it is a mistake we all have come to accept in common usage. The rabbis explain:
The original name for the people we now call Jews was Hebrews. The word "Hebrew" [in Hebrew, "Ivri" ] is first used in the Torah to describe Abraham (Gen. 14:13). The word is apparently derived from the name Eber, one of Abraham's ancestors. Another tradition teaches that the word comes from the word "eyver," which means "the other side," referring to the fact that Abraham came from the other side of the Euphrates, or referring to the fact Abraham was separated from the other nations morally and spiritually.
Another name used for the people is Children of Israel or Israelites, which refers to the fact that the people are descendants of Jacob, who was also called Israel.
[in Hebrew, "Yehudi"] is derived from the name Judah, which was the name of one of Jacob's twelve sons. Judah was the ancestor of one of the tribes of Israel, which was named after him. Likewise, the word Judaism literally means "Judah-ism," that is, the religion of the Yehudim. Other sources, however, say that the word "Yehudim" means "People of G-d," because the first three letters of "Yehudah" are the same as the first three letters of G-d's four-letter name.
Originally, the term Yehudi referred specifically to members of the tribe of Judah, as distinguished from the other tribes of Israel. However, after the death of King Solomon, the nation of Israel was split into two kingdoms: the kingdom of Judah and the kingdom of Israel (I Kings 12; II Chronicles 10). After that time, the word Yehudi could properly be used to describe anyone from the kingdom of Judah, which included the tribes of Judah, Benjamin and Levi, as well as scattered settlements from other tribes. The most obvious biblical example of this usage is in Esther 2:5, where Mordecai is referred to as both a Yehudi and a member of the tribe of Benjamin.
In the 6th century B.C.E., the kingdom of Israel was conquered by Assyria and the ten tribes were exiled from the land (II Kings 17), leaving only the tribes in the kingdom of Judah remaining to carry on Abraham's heritage. These people of the kingdom of Judah were generally known to themselves and to other nations as Yehudim (Jews), and that name continues to be used today.
In common speech, the word "Jew" is used to refer to all of the physical and spiritual descendants of Jacob/Israel, as well as to the patriarchs Abraham and Isaac and their wives, and the word "Judaism" is used to refer to their beliefs. Technically, this usage is inaccurate, just as it is technically inaccurate to use the word "Indian" to refer to the original inhabitants of the Americas. However, this technically inaccurate usage is common both within the Jewish community and outside of it.