No, we do not. For the simple reason that writing systems were only invented ~3200 BC, prior to that we have symbols but we don't know what they mean, and can't pronounce them. In contrast, people were living in cities (as opposed to camps or caves) as early as 8000 BC (see the site of Çatalhöyük in Turkey for example).

As I have answered in a previous thread, the oldest name we know is actually that of Iry-Hor, a king of Egypt in the Predynastic period, who reigned during the 32nd century BC.

So basically, we don't know any caveman names because cavemen were long gone and people had been living in cities and farming and herding for thousands of years by the time writing was invented.

Only in the uninteresting sense that there have been cave dwellers in to modern times in places such as Matera. That particular settlement goes back 9000 years, so well back into the Neolithic , so while of course the language has changed, it's not cheating too much to describe modern inhabitants as "cavemen".