Sigrid, Signild, Signe
But Sibyl Edith might be a bit too Downton Abbey...
Cybill Edith might be a little less Downton Abbey...
Savanna and Sabrina are kind of close close, but I think it's okay.
You could also just go with Zibby Edith...
I could sort of see Zibby for Thisbe, Thisbe uses all the letters of Edith except the D (and adds an S obviously).
Also somewhat for any of the names that are plays on/sounds like with Zephyr. (Zephyra, Zephyrine, Zafira, etc.). By extension, maybe Seraphina and Sephora?
Someone else mentioned Zenobia earlier. Edith Wharton, the author, wrote with a character named Zenobia, so there's a double tie to the name. Not the most positive character though (also not the wort, kind of complex).
Nameberry's entry on Edith has several interesting international variations, however none of them really help you out in terms of getting to a Z or a B in the middle name.
Zillah and Zipporah...
Zada (I think pronounced Zah-dah) is an Arabic Z name sharing some of the meaning ("prosperous") of Edith ("prosperous in war").
Ottilie (and its variants) shares Edith's meaning nearly to a T ('prosperous in battle'). I'm not sure it helps you to Zibby, although Tillie --> Tibby --> Zibby maybe?
Zara is another Z name possibility for Zibby.
I like the idea of deriving the nickname from the first/middle combo..
I like Sabrina, Sabina, and Sabine, but you're right. It's a little close to older sister Savanna.. :(
Congratulations on your new addition! I, agree Zibby is a very cute nn. You mentioned you liked Elizabeth, and a possible nn for this is Libby. I have paired Libby with some 'Z' names, so theoretically you could get Zibby.
Zayda Libby, Zara Libby, Zaydie Libby, Zanna Libby, Zaylia Libby, Zia Libby, Zilla Libby, Zoe Libby and Zora Libby.
Some names I think pair well with mn Edith - Felicity Edith and Matilda Edith. Piper Edith, Marlowe Edith, Harper Edith, Scarlett Edith, Vivienne Edith, Imogen Edith, Liliana Edith, Quinn Edith, and Molly Edith. Not sure if you could get Zibby from these, though!
Wishing you all the very best in your naming journey.
Zelda maybe? I think any Z name with a B middle name could work because then you would have Z.B. which could be pronounced "Zibby."