I think it would be an awful idea... The Avatar was awesome because the World(Fictional World but still the world) was at risk. Someone spending 20plus Episodes on trying to find their Mother seems more like a P.I show. Unless Aang, Katara and Comical Relief guy goes with him the show would be... Lets just say boring.
Doesn't Aang have more Important things to do then trying to find Zuko's mom?