Is there a specific place in the Prophet's Mosque where Lady Aisha secretly told her nephew Az-Zubayr that the Prophet, peace and blessings be upon him, used to pray there, and that whoever prays there will be forgiven by Allah? What is the name of this place or pillar?

Al-Zubayr is not the nephew of 'Aishah; rather, his children are the children of her sister. The mentioned story about 'Aishah has not been narrated. All that has been reported is what Al-Hafiz Ibn Hajar transmitted regarding the hadith of Salamah ibn al-Akwa' praying by a pillar in the Rawdah. Ibn Hajar stated that this pillar is the middle one in the Rawdah and is known as the Pillar of the Muhajirin. It was narrated from 'Aishah that she said, "If people knew it, they would contend over it with arrows," and that she confided this to Ibn al-Zubayr, so he used to pray there frequently. This was mentioned in Ibn al-Najjar's Tarikh al-Madinah, and he added that the Muhajirin from Quraysh used to gather there.