milk, suckle, breasts
嬭 centers on the act of a mother feeding an infant: providing milk, suckling, and by extension the breasts that produce milk.
嬭 combines 女 (woman) with 爾, which likely contributes the sound and may relate to the idea of closeness or intimacy, fitting the mother-child nursing context. The exact historical development is uncertain.
A woman 女 beside 爾, which resembles a child reaching out. Picture a mother holding her infant close to her breast to suckle milk.
For ダイ, imagine a mother nursing her baby while saying 'die' (as in 'that's my little die-die') — a playful invented sound for a nursing nickname: die -> ダイ.