It's kind of a tricky title, we know. Well, a and for are pretty straightforward, and once you start reading the story, you figure out what a Leibowitz is pretty fast. But canticle requires some digging.
A canticle is a hymn or song of praise. Although any song of praise can be called a canticle for artistic purposes, the name is mostly reserved for songs of praise taken directly from the Bible.
For example, the Benedictus (or the Canticle of Zachary) takes its lyrics from Luke 1:68-79. But A Canticle for Leibowitz doesn't come from the Bible. It's science fiction.
So what is Mr. Miller doing by throwing the word canticle in his title?
Canticle in a Can
Consider this: canticles are songs, and so they have a rhythm to them. In the same way, the events of A Canticle are set to the rhythm of history. Our history went from Early Middle Ages to Renaissance to Modern Era, and so does the fictional future history of the novel.
The people maybe different, and the events given different names, but the underlying beat remains unchanged.
Canticles teach religious heritage as well, often by snatching their lyrics right out of the Bible. So when you learn a canticle, you learn the teachings of the Bible. A Canticle plays with this nature of the canticle by blending religious heritage with scientific heritage, and contemporary issues.
You can peep this mélange in the novel's very own made-up canticle, a "Canticle of the Brethren of the Order of Leibowitz":
V: Lucifer is fallen.
R: Kyrie eleison.
V: Lucifer is fallen.
R: Christe eleison.
V: Lucifer is fallen.
R: Kyrie eleison, eleison imas! (24.25)
The religious and the scientific heritage together here. The Lucifer of the verse signifies the atomic bomb, and the response of the choir is for God's mercy (source). This canticle also sounds very much like something written during the Cold War, when nuclear war seemed a very real prospect.
The fear was so real that schools had drills for just such a war, complete with cartoon turtle spokesmen and everything. And again, notice the repetition featured in this canticle; this repetition parallels our own history, as we discussed earlier.
So that's that. Now, dear Shmooper, you know why Mr. Miller put the canticle in A Canticle for Leibowtiz.