Prior to the 1960s, the theory was variously known by the names of Explosion, Creation, Evolution, Catastrophe, and even "Fireworks" theory. Specific variations/elaborations were given the name of their theorists (e.g. Gamow-Alpher-Herman Theory).

The term "Big Bang" was coined by Fred Hoyle in 1950 as a pejorative, dismissive term (he favored a Steady State model, whereby the universe was infinitely old), deliberately meant to make it seem improbable. He didn't capitalize it — it was just how he described the theory:

...the whole of the matter in the Universe was created in one big bang at a particular time in the remote past. On scientific grounds this big bang assumption is much the less palatable of the two. For it is an irrational process that cannot be described in scientific terms. Continuous creation, on the other hand, can be represented by precise mathematical equations whose consequences can be worked out and compared with observation. On philosophical grounds too I cannot see any good reason for preferring the big bang idea.

Some of those involved with these discussions (like George Gamow) really disliked the term because it associated it (in their minds) with nuclear explosions (a "hot" topic at the time). Though some of them also embraced that as a metaphor, too, both as a popular explanation and as a way to understand the high temperatures involved.

By the 1960s, however, the term had stuck and lost its pejorative association. There is, to be sure, a measure of irony of an opponent having coined the “successful” term; this sometimes happens, where the insult becomes re-appropriated and the original negative connotation is forgotten.

There is much more that can be said about the theory's reception and cultural connotations (the big issue was not so much its association with explosions, but the deus ex machina nature of it that made a lot of people think it is was essentially religion dressed up as science), separate from the name.

The standard reference for this history is Helge Kragh, Cosmology and Controversy: The Historical Development of Two Theories of the Universe (Princeton University Press, 1996). I have also benefitted from the recent book of my good friend (and my discussions with him): Nasser Zakariya, A Final Story: Science, Myth, and Beginnings (University of Chicago, 2017), which has a lot about Gamow’s views in it (Gamow was not only one of the key developers of the theory in the 1940s, but one of its chief popularizers; Zakariya's book is about the kinds of grand narratives that scientists tell). The Hoyle quote comes from Fred Hoyle, The Nature of the Universe (Harper & Brothers, 1950), 124.