Looking back at the 1881 census for Eliza and her children Denis and Mary, there is a Patrick Coil age 27, laborer, born Ireland in the household. I don't have the original to see the handwriting but I am thinking it is Patrick McIlroy. He is 45 on the 1901 census.
Peggy
Here is a snippet from the 1881 entry from FindmyPast. Intentionally written as Coil, but you are right in concluding Patrick McElroy being on the Census. Although it appears they have his age incorrectly recorded and the spelling of his last name
Findmypast truncates the listing to 5 rows is more are on the Census. Which I previously missed opening up the additional entries. Here's the 1881 Census again with all entries including Patrick McElroy. Plus the 1891 Census listings.
1881 Census
Eliza Hurl Head Widow Female 34 1847 - Ireland
Dennis Hurl Son Single Male 11 1870 - Newcastle on Tyne, Northumberland, England
Mary Hurl Daughter Single Female 5 1876 - Newcastle on Tyne, Northumberland, England
Patrick Coil Lodger Single Male 27 1854 Laborer Ireland
Michael Mc Guign Lodger Single Male 24 1857 Laborer Ireland
John Crilly Lodger Single Male 40 1841 Laborer Ireland
Patrick Mackleroy Lodger Single Male 39 1842 Laborer Ireland
1891 Census
Eliza Hurle Head Widow Female 44 1847 - Newcastle
Denis Hurle Son Single Male 21 1870 - Ireland
Patrick McIvy Boarder Single Male 34 1857 General laborer Ireland
John Dixon Boarder Single Male 40 1851 General laborer Ireland
John Smith Boarder Single Male 35 1856 General laborer Ireland
Peter Michael Boarder Single Male 24 1867 General laborer Ireland
Patrick Inn Boarder Single Male 24 1867 General laborer Ireland
Charles Quin Boarder Single Male 24 1867 General labourer Ireland
Patrick McBride Boarder Single Male 24 1867 General labourer Ireland
Michael Downey Boarder Single Male 19 1872 General labourer Ireland