I would suggest Liz, Lizzie, Beth, but my favorite is definitely Libby.
i would suggest Liz Lizzie Beth too but maybe just eliza (e-lie-za) that's how to pronounce it.
Ellie, Ella, Lizzie
Her Majesty Queen Elizabeth does not have nicknames.
Good Queen Bess and the Virgin Queen are two amongst many.
Good Queen Bess, Gloriana, The Virgin Queen
Elizabeth Imus goes by Elizabeth Imus-Zero, and Elizabeth Zero.
Elizabeth Arbilon goes by Noalyn.
Elizabeth Schell goes by Lizzie.
Elizabeth Sampson goes by Sam.
Elizabeth Powers goes by Eliza.
Elizabeth Novotny goes by Liz.
Elizabeth Reiners goes by Beth.
Elizabeth Spaetzel goes by Liz.