Ivrit is the Hebrew language. Until the 20th century, however, Hebrew was called lashon hakodesh, the holy tongue, the language of prayer and Torah study.
The early Zionists liked to refer to themselves as Ivrim, Hebrews, to distinguish themselves from the stereotypical humiliated diaspora Jew, an image which embarrassed them. Vladimir Jabotinsky (1880-1940) was typical in declaring, The Yid is trodden upon and easily frightened, therefore the Hebrew ought to be proud and independent. The revived Hebrew language of the Ivrim was, naturally, Ivrit.
The first person to be called an Ivri was Abraham (Genesis 14:13). The commentators disagree why. Rashi says it was because he was from meever, or beyond, the river. Radak claims it alludes to Abraham being a spiritual descendant of Ever, the monotheistic great-grandson of Noah. When Potiphars wife accuses Joseph of trying to rape her she calls him that eved Ivri, or Hebrew slave. The Egyptians would not eat in the same room as Josephs brothers because they found it abominable to fraternise with Ivrim.
In each of these cases, Ivri is used to refer to the biblical characters ethnic origins. It was natural, then, for secular Zionists to borrow the word to describe a national identity cut off from its religious roots.