The story of the name, from the sources
Aisha comes from aysh, life. Ibn Faris says the root means life, and lasting. And in Lisan al-Arab, al-aysh is life. So Aisha is the one full of life.
“a root that means life, and lasting”
“al-aysh is life”
Maqayis al-Lugha · vol. 4, p. 194 · Lisan al-Arab · vol. 6, p. 321
And al-Zabidi has a surprise in Taj al-Arus: Aisha was a name for men and for women. One of those men owned a well near Medina, called Aisha's Well.
“Aisha: a name for men and for women”
One of those men owned “Aisha’s Well”, near Medina
Taj al-Arus · vol. 17, pp. 282–287
Its most famous bearer is the Mother of the Believers, Aisha, daughter of Abu Bakr, may God be pleased with them both. Masruq said: I saw the senior Companions of the Messenger of God, peace be upon him, asking her about the laws of inheritance.
Aisha, Mother of the Believers
“I saw the senior Companions of the Messenger ﷺ asking her”
Masruq
Ibn Sa’d, al-Tabaqat · vol. 2, p. 286 · vol. 8, p. 53 · al-Isaba · vol. 8, p. 233
Her nephew Urwa ibn al-Zubayr said: I never saw anyone who knew more of law, or medicine, or poetry, than Aisha. And: whatever happened to her, she had a poem for it.
“I never saw anyone who knew more of law, medicine or poetry than Aisha”
‘Urwa ibn al-Zubayr
“Whatever happened to her, she had a poem for it”
al-Isaba · vol. 8, p. 233 · vol. 1, p. 56