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