It was about the national identity, not religious reasons. More religious segments of Judaism were historically (and often still are) skeptical of Zionism in general, of which the Hebrew revival was a part.

First, some linguistic background. Hebrew went extinct by the 2nd century. But it remained in use in Jewish communities as the language of the liturgy, Jewish texts, and religious literature. Synagogues still generally used Hebrew biblical texts, religious learners still learned Jewish texts, Rabbis still wrote Hebrew commentaries. Jews were generally literate in Hebrew script, and used it for writing non-Hebrew languages in many cases.

The revival of Hebrew is an interesting linguistic topic. The father of Hebrew is Eliezer Ben-Yehudah. But the process began a bit before. During the 19th century, the Jewish literary revival involved making Jewish works of "high" literature, in both Hebrew and Yiddish. One product of this was a body of literature in Eastern Yiddish (Tevye the Milkman was a product of this, which of course became Fiddler on the Roof). Another was new literary writing in Hebrew, by authors like Chayim Nachman Bialik.

It's with this backdrop that we get a critically important, but sadly forgotten, Zionist named Ahad Ha'am (that was his penname--his actual name was Asher Grinsberg. Ahad Ha'am means "one of the nation"). There were several strands of Zionism in the late 19th century. His was sometimes called "cultural Zionism". He saw the establishment of a Jewish state as the ultimate expression of this Jewish cultural revival. Just as writing literature would revitalize Jewish writing, a Jewish state would revitalize all aspects of Jewish life. He saw Zionists like Herzl as being sort of half-assed. They wanted to help Jews, yes, but they didn't want to really make Jews a more vibrant nation. In essence, he saw Herzl et al as trying to create a "state of Jews", while he wanted a Jewish state. He didn't think it was practical to just get Jews to migrate to Palestine--you'd need a cultural revival to get people interested in such a national enterprise

And he wrote these objections in Hebrew. One of his bigger essays, זה לא דרך (Hebrew for "this is not the way"), where he outlined his objections. Trying to build a state out of nothing is pointless--first, Jews have to be inspired by a revitalization of Jewish culture.

Around this time Eliezer Ben-Yehudah started working to revive Hebrew. He did a few things to this end. He wrote a Hebrew-language newspaper based in Jerusalem. He raised his son as a native monolingual Hebrew speaker. He wrote a Hebrew dictionary, including a number of coined words for new concepts.

To address your initial question, there were religious objections to this. Hebrew was, in Jewish parlance, lashon kodesh/loshn koshesh, the Holy Tongue. It had a religious significance. And many objected to its use in daily affairs. It was a particular point of controversy for Eliezer Ben-Yehudah in Jerusalem, which had a large religious population.

With this intellectual and ideological backing, combined with someone doing the legwork, Hebrew slowly started to gain ground. In the early 1900s Zionists incorporated Revisionist Zionist lingo into the language arguement. They saw Yiddish as an exilic language, which was to be eschewed for the real Jewish language, Hebrew. This active objection to other languages, combined with the ideological advantage of a Jewish revival it provided, gave it strength.

Perhaps the solidifying moment was in the establishment of the Technion. It's a scientific univerisity in Haifa. There were intense debates about what its language should be. On one hand, German was the language of scientific discourse at the time. On the other, it was a institution in the Jewish statehood enterprise, and it should use the Jewish language. Eventually, Hebrew won out.

So propelled by a couple pieces of Zionist ideology, Hebrew managed to become the language of what would become Israel.

Edited to remove typo