as you are there, heres my offering
not straight forward
Second Ballyeaston Presbyterian Church [the church with burial  records 1842-9 and 1901-18]
19 Trenchill Rd, Ballyeaston, Ballyclare BT39 9SJ
ie West of Ballyeaston village
400 yards away from 
First Ballyeaston Presbyterian Church
Lower ballyboley Road,  Ballyeaston, BT39 9SH
ie East of Ballyeaston village
which in turn is going 400 yards further eastwards is the start of Ballyalbanagh Road
1901 Census from 
http://www.census.nationalarchives.ieYour Eliza is not Lizzie Davidson living in 1901 at 41 in Ballyalbanagh (Ballycor, Antrim)
Your Eliza is not Elizabeth Davidson living in 1901 at 34 in Ballyeaston Town [ballyclare]
the most likely person in 1901 census is
2 in Ballyeaston (Ballyclare, Antrim)
Davison   Eliza   75   Female   Head of Family   Presbyterian   Co Antrim   -   Read and write   Irish and English   Widow   -
Davison   Robert   25   Male   Grand Son   Presbyterian   Co Antrim   Farm Servant   Read and write   Irish and English   Not Married   -
Although name entered for each as Davison I think in a different hand, the final signature is E liza Davidson.  Whose property included a fowl house
Griffiths valuation continuation books 
https://apps.proni.gov.uk/ for Ballyeaston see VAL/12/B/1/4A (1864 - 1866)
John Davidson has crossed out entry for 3a John Davidson crossed out, replaced with James Gilmore/Gilmour has 5 acres no building
time reasons left out some books
Griffiths valuation continuation books for Ballyeaston see 
VAL/12/B/1/4E (1897 - 1904)p164
12 Wm Thos Davidson has house valued £1 pa
15Ba Robert Davidson living with note change 1898 building valuation £1 ten shillings pa looks like left to son by landlord rather than widow
VAL/12/B/1/4E (1897 - 1904)p164
17 Brown Davidson has6 acres no bldg
Griffiths valuation continuation books 
https://apps.proni.gov.uk/ for Ballyeaston  Village of Ballyeaston 
Ballyeaston  Village of Ballyeaston 
Brown Davidson Lic’d  under
VAL/12/B/1/4E (1897 - 1904) p175
44.17 John Davidson Ho Office  
44.18 Archibald Davidson Ho Office and garden
VAL/12/B/1/4E 
44.17   15Ba Robert Davidson crossed out Davidson in 1907 Ho Office  
44.18 Archibald Davidson Ho Office and garden
that lead me back to the griffith primary valuation
Davidson   John   Ballyeaston       Ballycor   Antrim
has 3 acres land at 3A. sadly the particular map used has no map refs for the townland , the original might be seen at Ballymena Hdgs library or the current valuation office, Belfast
Davidson   Hugh   Ballyalbanagh   Ballycor   Antrim has  map ref 66   21 acres and a house valued £1 five shillings. No other houses are visible, as unexpected 
 I think this map ref works showing a the top end of 2nd unnamed road going northwards of the Lower Ballyboley road at 50 lower Ballyboley Road. in an easterly direction spiral northwards showing a large barn sitting over a house
going to a satellite view the google map direction is ttps://www.google.com/maps/dir/54.7803362,-5.9704554/Lower+Ballyboley+Rd,+Ballyclare+BT39+9SL/@54.7891346,-5.9736499,43m/data=!3m1!1e3!4m9!4m8!1m0!1m5!1m1!1s0x4861a576996a875b:0x5e208b266bfa7f3b!2m2!1d-5.9975339!2d54.7742634!3e0
Also ref 57Bc John Davidson is living in a modest house valued at 5 shillings pa. I can find 57B but not 57Bc the townland suddenly is constricted just below its most northerly neck where a place name exists Whitefordstown this a mile and a quarter, northwards, the road divides the north westroad immediately becomes the s w part of John Davidson little house area.   His homestead that bldg. marked with a c might be indistinct within the constricted neck small bulge  above the constricted neck
see 
http://griffiths.askaboutireland.ie/gv4/single_layer/i8.php?lat=&longt=&dum=0&sheet=39,45&mysession=2605855667868&info=&place=&county=ANTRIM&placename=BALLYALBANAGH&parish=BALLYCOR&country=Ireland&union=&barony=ANTRIM,%20UPPER I close with a peep at the Tithes Applotment book Tithes Applotment Books 
1823-1837 to find no listing appears for Ballycor   The relevant original books are in the 
Public Record Office of Northern Ireland (PRONI) Belfast it would be worth checking no listing exists.