I'm sorry I don't have any info on Eliza Briggs directly, but I do have a direct link with a Robert Briggs in Tuxford Nottinghmanshire. I have no middle name for him nor do I have any info about his siblings, but I do have info that his wife was Elizabeth Hill Whittington, and that they had a son in 1844 - Joseph Briggs - while still in Tuxford. Whether Joseph was an only child or not I don't know. But this would make Robert Briggs about the right age to be Eliza's father, or since you say her father's name might be George, he might her uncle! (I know it's tugging at straws but is it possible that Eliza was named after her aunt?). I found some reference in the Nottinghamshire Archives (searched on www.a2a.org.uk) to the existence of a copy of Robert Briggs' will and title of trustees dated about 1846. If we could find a copy of that we might be able to determine if Robert had any more kids and/or if Robert had any brothers possibly named George! I'll see what I can lay hold of in some of the Family History centres around here.
I hope this is at least a small help. And PS I live in NZ too - Hamilton! Cheers, Jeremy