The name "Big Bang theory" was coined by Sir Fred Hoyle. What did scientists originally call the Big Bang model?
Lemaitre, who proposed the first version of "Big Bang" in 1927, called it Primeval Atom hypothesis since late 1930s, notably in the 1950 book of this name. However, it was rather different from the subsequently adopted version, and singularity cosmologies did not gain much traction until 1950s to get a common label. By then, when they came to be viewed as a credible alternative, Hoyle's "big bang" nickname from 1949 stuck. On Lemaitre's motivations see Kragh-Lambert, The Context of Discovery: Lemaître and the Origin of the Primeval-Atom Universe
"It is generally acknowledged that the priority of the Big Bang idea belongs to Lemaître. As the eminent cosmologist Jim Peebles wrote in Physical Cosmology, the first textbook of the hot Big Bang standard model: ‘According to the usual criterion for establishing credit for scientific discoveries Lemaître deserves to be called the “Father of the Big Bang Cosmology”.’... The primeval atom did certainly not have factual status in the 1930s, and today we are convinced that what Lemaître ‘discovered’ has never existed. Our present picture of the very early universe has almost nothing in common with the gigantic atomic nucleus that Lemaître envisaged. Nonetheless... Lemaître may be said to have discovered the Big Bang in the form of the now abandoned hypothesis of the primeval atom.
[...] He did not care to distinguish between the terms ‘nucleus’ and ‘atom’, for the atomic number of the primeval atom and its decay products was so large that it made the distinction illusory. He later spoke of the primeval atom as a kind of isotope of the neutron, a metaphorical terminology he first adopted in an unpublished manuscript from the late 1930s:
"This hypothesis [of the primeval atom] is attractive only if we can think of the atom as a quantum unit. Then, we start really from ideally simple conditions. We have surely to think of the primeval atom as a base nucleus without surrounding electrons, an isotope of the neutron.""
Georges Lemaître called it “hypothesis of the primeval atom” in 1931. See Georges Lemaître, Father of the Big Bang, which is an excerpt from Cosmic Horizons: Astronomy at the Cutting Edge, edited by Steven Soter and Neil deGrasse Tyson. He also called it “The Beginning of the World”.
Lemaître model was a "cold matter" one. Our present model of "hot" Big Bang can be traced to the "$\alpha\beta\gamma$" model by Alpher, Bethe and Gamow in 1948.
It only became generally accepted after the discovery of the cosmic microwave background radiation by Penzias and Wilson in 1964.