I'm no fan of Elizabeth and Eliza is worse. It's currently trendy and sounds kind of yuppie, but at the same time it's always struck me as a servant's name.
Not trendy in just numbers; I think it's going to be one of those filter-down-from-the-elite kind of names, like Lucy and Emma. It just carries such a strong vibe of horses and nannies and "nurseries" not in the sense of a baby's bedroom.