First you have to find Liza's maiden name.
Your best bet is a birth certificate for one of Liza's children or her marriage certificate to James. She was only about 17 when she had James in Liverpool in 1848 or 1849. So chances are that she got married in Liverpool at about the same time. There was an Eliza Gelling born in Liverpool in the June quarter of 1860 (Volume 8b page 223) and an Elizabeth in the June quarter of 1859 (Volume 8 b page 158). Either is a possible match for James's and William's sister. The first is more likely.
With this information you can apply for her birth certificate on line with a credit card to the Public Records Office. http://www.familyrecords.gov.uk/frc/research/bmdmain.htmIt costs about 8 pounds and then you have to hope that you have the right Eliza. The birth certificate should give the mother's maiden name. Freebmd is putting post 1837 births, marriages and deaths online. So far ithe work is about 80% complete for 1847-1850. When i is complete you should be able to get the marriage info for those years. A copy of the marriage certificate will give Eliza's father's name and occupation. If you cannot wait some professional researchers will do a search for about 20 pounds.
The Public Records Office in Dublin was detoyed by fire in 1922 during the Irish Civil War. So even if you find Eliza's father, you may have trouble going further back in Ireland.
However given her young age, her whole family may have moved to Liverpool.