Abbaye saying that he felt so confident in answering questions once, he was like ben Azzai in the markets of Tiveria: Kiddushin 20a
Rava telling his students, on a day when he felt good, "Today I can answer questions like ben Azzai did in the markets of Tiveria!": Eruvin 29a
Whether or not ben Azzai ever married: Sotah 4b
Rabbi Akiva's daughter did for Ben Azzai as Rabbi Akiva's wife did for him, sending him off to study in yeshiva for many years, emulating her mother: Ketuvot 63a
Nachum Ish Gam Zu whispered a leniency to Rabbi Akiva, who whispered it to Ben Azzai, for fear that if it spread it might lead to improper practices. It's not clear whether Ben Azzai preserved secrecy, or not: Berachot 22a
Ben Azzai reporting a halachah taught on the day Rabbi Elazar ben Azaryah ascended to lead the Study Hall: Zevachim 11b