I say the name
Ada as AY-da. I like it as a full name.
I don't feel like it makes enough sense as a nickname for
Adelaide.
I could see Aidi(e) "Ay-dee" as a nick for
Adelaide, because the Aid syllable is emphasized, but adding an A to that seems artificial to me, esp. since
Ada is a name.
Ada "add-a" for
Adelaide would not work very well either, IMO, because
Addie exists, and it just seems weird, like calling
Abigail (even Abagail) "Aba".
- mirfak