There is no real evidence that cavemen had names, or, at least, what we consider valid as names nowadays.
However they would have to, somehow, get the attention of one another eventually and assuming language, if it existed at all, was not very sophisticated, their "names" would have been something somewhat short and based on guttural sounds which wouldn't require too much thought.
Some authors, like Jean Auel, have written stories (The Clan of The Cave Bear) set in the primitive era and when faced with the problem of coming up with names for its characters decided to stick to exactly that, basic names based on distinctive short sounds one imagines they would call each other to get their attention.
Some examples of this are:
Brun, Talut, Durc, Jep, Yef, Ubi and Thak - For men
Ayla, Ika, Latie, Muta, Rake, Oba and Boka - For women
Once again this is based on works of fiction as there are no known records of any written documents produced then.
It's hard to imagine they would be too concerned with this after spending most of their time hunting for food and trying not to become food themselves.