Hisham bin Urwah narrates from his father that Sayyidah Aishah (may Allah be pleased with her) said to the Messenger of Allah (peace and blessings be upon him): "All your (peace and blessings be upon him) wives have a kunyah except me." He (peace and blessings be upon him) said: "Take your kunyah after your son Abdullah bin al-Zubayr (may Allah be pleased with them both)." So the kunyah of Sayyidah Aishah (may Allah be pleased with her) was Umm Abdullah until she passed away.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الضحايا / 19334
Abbad bin Hamzah bin Abdullah bin al-Zubayr narrates from Sayyidah Aishah (may Allah be pleased with her) that she said: O Messenger of Allah (peace and blessings be upon him), why do you (peace and blessings be upon him) not give me a kunyah, when all your (peace and blessings be upon him) wives have kunyahs? He (peace and blessings be upon him) said: "Take it after your son Abdullah." So her kunyah was Umm Abdullah.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الضحايا / 19335