Looks like the 1901 census is correct for Eliza!
I think you are onto something with #9 Boyher - My mum thought Johns father was called William and a William is at this address.
http://www.census.nationalarchives.ie/pages/1901/Monaghan/Dawsongrove/Boyher/1633517/So William could actually be Johns Brother due to the age! There's also a Martha. Now my mum said Martha was a child to John and Miss Rainee but she could actually be Williams wife. But then the other Children's names don't tie up with anyone we know. In the 1911 Census William and Martha are living at #6 Rockcorry, with a daughter Elizabeth (she's not listed in 1901). Again a name my mum said was a child to John and Miss Rainee
http://www.census.nationalarchives.ie/pages/1911/Monaghan/Dawsongrove/Rockcorry_Town/806441/What would be interesting is the wedding record for William and Martha to see if Williams father is also Thomas which would suggest William and John are brothers. Also to figure out Marthas maiden name.. I wonder if thats similar to Rainee? Does anyone have access to that record?
But the 1902 wedding record for John and Eliza said John was a widower so there must be another marriage record somewhere!
Not sure about John Patterson at #25 in 1901 as it says he's 45, whereas in 1902 he says he's 50 when he got married but as Eliza is living at #22 then that could be how they met!
Not sure about the 1901 census link you sent.
Rooney and Rainee sound very similar!
LINK to Eliza Jane Gardner on 1901 census at #22 in Boyher (Dawsongrove, Monaghan)
There is a William Patterson & family at #9 and a John Patterson at #25 in Boyher (Dawsongrove, Monaghan)
I wonder is this your John although he is shown as not married if you look at the original return there is a mismatch of columns and he could have been the widower - LINK
Also, with wrong father's name this may be unrelated
John L PATTERSON aged 27 of Riverside (F Isack)
married Feb 16 1885 in Dungannon, Co. Tyrone
Jane ROONEY aged 25 of Killymorgan (F William)