The evidence that supports the idea that Cosmic Microwave Background (CMB) radiation is a remnant of the Big Bang includes "Its temperature fluctuates greatly." These fluctuations, known as anisotropies, provide important information about the early universe's density variations and the formation of large-scale structures.
However, it's important to note that the statement about "Its temperature is uniform" is also characteristic of the CMB radiation, as it has an overall uniform temperature of about 2.7 K, which reflects the cooling of the universe since the Big Bang.
In this context, the most direct evidence supporting the Big Bang theory concerns the observations of the fluctuations in temperature of the CMB, as they provide crucial insights into the early universe's conditions.