Marriage Certificates - there was no 'rule' about what should go into the father's name and occupation boxes. Sometimes it was the 'father figure' and not the biological father. And what do you put if the father has already died? Sometimes you find nothing, sometimes the boxes are filled in as if he was alive. My OH's father had already died when we were married, so we put his full name (deceased) and the occupation he had when he was alive. That 'should' be clear for anybody coming along in years to come, but of course doesn't help in the past.
In Eliza's case, was there a baptism? Just occasionally you get the father's name, even if subsequently the two didn't marry.