Hi Elwyn and Patricia,
I was overwhelmed by your speedy and detailed responses which I was not expecting - thank you! I did not have the information previously, but yesterday obtained Eliza's death certificate which stated that a Presbyterian minister presided at her funeral, so a visit to PRONI might assist with any baptism records. Thanks for the tip about obtaining a researcher; I live in Melbourne, Australia, and am not planning a trip to Europe in the near future, so I will give that some thought.
I checked out the link to the Catholic records you so kindly provided, as I have another Irish female ancestor - Mary Neil, a Catholic convict from Limerick about the same age as Eliza. The records are a challenge to decipher online and I don't have any information about Mary's parents, so that might be another task for a future trip.
I had a look at the Stewarts who were on the Roman Emperor with Eliza. They came from towns that seemed a good distance from Dungannon, Coleraine and Newry. Could the sisters have been split up like that back then? I'm guessing too that Stewart was a very common surname?
Thanks for the information that Eliza might have been illegitimate - I did wonder. She herself lived de facto for 19 years and had eight children illegitimately with her second husband before they married, presumably after word reached her that her first husband had died.
My other Irish ancestors were vague about where in Ireland they were from, but Eliza particularly mentioned Dungannon Parish on her marriage certificate, so I think she must have felt strong ties to her birth place. You must too to be volunteering in this role. Thanks again for the help. Regards, Julie Brown.