There is no special name for someone who studies Hebrew.
The ancient Israelites (before our time in Egypt) were apparently a nomadic tribe whose name is usually transliterated as "Ibaru", possibly originally meaning "the ones from across the river." We were called Hebrews by the Egyptians, and -- at least technically -- did not become Israelites until we were established in the land of Canaan and had given up our nomadic traditions. The everyday language of the Israelites was Aramaic. The traditional history of the Israelites -- what came to be known as the Hebrew Bible or the Old Testament -- was in Hebrew. Hebrew was therefore used primarily for religious purposes. So ... today, Jews are occasionally called "Hebrews", but Hebrew is actually the name of a language, either traditional Biblical Hebrew or the modern version of it currently spoken in Israel.