There is also this from the 1881 census, which is an Elizabeth Barnes, a widow born around 1848 in Ireland with her son William born in Liverpool around 1873, which fits with the 1893 death:
https://familysearch.org/ark:/61903/1:1:XQD8-X8R
I don't have her in the 1891 census, but the 1893 death is looking more consistent. I'll get the certificate ordered later on.
Is this a possibility for Eliza a boarder in Liverpool
Her surname transcribed as Baucke and looks to read Bourke not Barnes

1891 RG12; Piece: 2912; Folio: 61; Page: 16
Eliza boarder widow age 49 Ireland Co. Wexford occ ?house woman
Looking to see if I can find an Eliza Bourke/Baucke on 1881 census but haven't come up with anything yet 
Yes, it is transcribed as Bourke here (she was a warehouse woman):
https://familysearch.org/ark:/61903/1:1:791C-M6ZLooking at the actual census page, it certainly looks like Bourke. It may be they misheard, but the age is also out. Norfolk Street is on the south side of the city centre down by the docks. So I'd put it down as a maybe.
I suppose it depends on when she died - it might be hers is the earlier death. We'll wait and see.
Do you have Eliza’s son William Barnes on 1891 census?
I found a couple of possibilities:
William Burnes aged 21 - the age is off (we'd be looking at 18-19) but he is a ship's fireman (very telling now we know his father's job, although he is listed as a ship's painter on his wedding certificate and in the 1911 census but you could imagine, given his father's fate, that he might want a safer job when he settled down) and he is living on Silvester Street which is almost opposite St Anthony's. He is lodging with three other stokers and there is a bracket around them saying... "seas"?, one of them appears to be a John Burnes but he is born in Scotland
so they aren't brothers [edit: rechecked, John Burnes is 40, so he could be his father]:
https://familysearch.org/ark:/61903/1:1:79K2-RN2William Barns aged 17 in the
Toxteth Park workhouse:
https://familysearch.org/ark:/61903/1:1:7SRW-66ZEither census result would explain why he wasn't living with his mother at this point, assuming she was still alive.
Possibly relevant is an 1890 conviction of a William Barnes aged 19, a ship's cook who stole a coat and a pair of socks from one Thomas Ryan and got 1 days imprisonment.