It's in my family tree several times as a full name, and it's grown on me too. I prefer it to
Elizabeth and most of the other nns for
Elizabeth, and in fact the only con I can think of is that people would probably assume it was short for
Elizabeth.
The family combos are
Eliza Rosetta,
Eliza Amelia,
Eliza Jane and
Eliza Florence. I also like
Eliza Lucy and
Eliza Susanna.