Some sources state that the name also means "brave, courageous", and interestingly, they use the word ვაჟკაცი (vazhk'atsi) to give that meaning with. "Vazhk'atsi" is a composite word that consists of Georgian ვაჟი (vazhi) meaning "son" and Georgian კაცი (k'atsi) meaning "man", so a more literal meaning of the word would be "manly man" or "manly son". This is reminiscent of the Germanic given name Carloman, which is composed in a similar manner.All in all, I suppose that one could say that while the given name Vazha may mean "son" in a literal sense, it can be considered to mean "brave, courageous" in a more general or figurative sense.Lastly, the popularity of Vazha as a given name started in honour of the celebrated Georgian poet Vazha-Pshavela (1861-1915), whose real name was Luka Razikashvili. Only the popularity, mind you: the first use of the name did not start with him. After all, the use of appellatives as given names is a common thing in the Kartvelian naming culture:"One group of first names is derived from appellatives, denoting: a person of both genders and of various ages (Geo. Gogona, Megr. Cira, ʒaba, Svan. Dina "girl"; Geo. ḳacia, Megr. ḳočia "man"), and relatives, as a rule, of the older generation (Geo. Dediḳa, mamia, papuna, biʒina. Their stems denote: mother, father, grandfather, uncle. The rich derivative capability of Kartvelian languages is extensively used for the formation of such names." ~~~ from page 891 of "Namenforschung/Name Studies/Les noms propres" at https://books.google.de/books?id=ck1CLzJq9vAC&pg=PA891 (in English)Also see:- https://imya.com/name/21924 (in Russian) - http://kids.ge/baby-name?id=417 (in Georgian) - https://web.archive.org/web/20140726005647/http://name.interes.ge/dreams.php?action=view&id=382&from=action=search|by=%E1%83%95 (in Georgian) - http://www.geogen.ge/ge/msearch/4309/ (in Georgian; scroll down to the entry for Vazha) - ვაჟკაცი (vazhk'atsi) meaning "brave, courageous": https://en.wiktionary.org/wiki/%E1%83%95%E1%83%90%E1%83%9F%E1%83%99%E1%83%90%E1%83%AA%E1%83%98 (in English) - Vazha-Pshavela (1861-1915): https://en.wikipedia.org/wiki/Vazha-Pshavela (in English)
Sources: https://www.names.org/n/vazha/about
https://forebears.io/x/forenames/vazha