Wiktionary has "From Old Georgian ძჱ (ʒē), ძეჲ (ʒey, “son”)." (IPA d͡ze).
Most of these surnames have the characteristics of dithematic names, rather than compounded names like
Anderson, except for გიორგაძე giorgaʒe from გიორგი giorgi (from Greek Georgios)+ძე son