Eliza is more annoying than Elizabeth. Elizabeth has been more or less steadily popular for forever, but Eliza was popular for a while and then faded out by the time 1900 rolled around, and now it's a yuppie/hipster favorite. Plus it sounds like a maid, whereas Elizabeth doesn't. And Elizabeth has all kinds of other nns to choose from but Eliza really doesn't.
Eliza Jane reminds me of the rhyme the kids sang in "Little Town on the Prairie" to make fun of their teacher, Eliza Jane Wilder. It went something like
It's lots of fun to go to school,
Laugh and grow fat is the only rule.
We laugh until we have a pain,
at Lazy Lousy Lizy Jane.
I think, therefore I judge.