Eleanor,
Griffiths Valuation (1859) lists 3 William Davisons in Drumrot. All were farmers. William junior had plots 2 to 4 (20 acres), William senior had plot 16 (25 acres) and there was a William (James) on 12a & 13 with 22 acres. Senior & junior may well be father & son, the other William is probably related too but more distantly.
https://www.askaboutireland.ie/griffith-valuation/index.xml?action=nameSearch
Going back in time, to the 1831 census we find William Davison (transcribed as Danison) with 5 folk in his family:
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Loughinshollin/Lissane/Drumrot/15/
Ellen D with 7 (plus a servant). Ellen was probably a widow, and dead by 1859. (Ellen & Eleanor are interchangeable names in Ireland).
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Loughinshollin/Lissane/Drumrot/16/
William jr with 5 in his family:
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Loughinshollin/Lissane/Drumrot/17/
Robert & George sharing a property, total 5
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Loughinshollin/Lissane/Drumrot/18/
Jane with 6. Again she's likely to be a widow and dead by 1859, since she’s not in Griffiths.
http://www.census.nationalarchives.ie/pages/1831/Londonderry/Loughinshollin/Lissane/Drumrot/19/
The enumerator numbered the properties 15 – 19, indicating they were all very close to each other, the inference being they were probably related.
Going back to the tithes in 1827, there were 4 farms there, 3 held by widows and 1 by William. Each has a name beside them to distinguish them, and is probably their late husband’s name. The tithes only listed those with land ie farmers, so the Robert & George in the 1831 census who are not in the tithes had some other profession without land eg farm labourer.
http://www.irishgenealogyhub.com/tyrone/tithe-applotment-books/parish-of-lissan.php
There’s a couple of possible deaths for 2 Williams registered in Magherafelt. One died 1870 aged 80 and the other 1865 aged 47. Neither death is viewable free yet so you would need to pay to view on the GRONI website (£2.50 a time). I don’t see a death for Eliza Davidson, so possibly she died before 1864.
http://www.irishgenealogyhub.com/tyrone/tithe-applotment-books/parish-of-lissan.php
You can use the Valuation Revision books to see who was the occupant of the various properties in Drumrot, post 1859:
https://www.nidirect.gov.uk/information-and-services/search-archives-online/valuation-revision-books
For example, plots 2 to 4 changed to what looks like Abraham Davison in 1872, then to George Shannon and in 1877 to Janet Davison.
2 Davison farms in Drumrot in 1901:
http://www.census.nationalarchives.ie/pages/1901/Londonderry/Moneyhaw/Drumrott_2/1534212/
http://www.census.nationalarchives.ie/pages/1901/Londonderry/Moneyhaw/Drumrot_1/1534291/
Coleraine Chronicle 15th Jan 1848 reported the death of Robert Davison aged 58, on 1.1.1848 at his residence in Drumrot.
Newry Telegraph 27th Feb 1844 mentions William Davidson of Drumrot in connection with the sale of a farm in the townland belonging to Widow Junk.
PRONI has a number of leases relating to the Davidson family of Drumrot, eg on in 1820 to Eleanor D (so a widow by that date) others to Jane and to William in 1820, one to Elizabeth in 1827; Andrew in 1854; George in 1854; William jr in 1854; William snr in 1854; Bargain & sale by William to George Shannon in 1862 mentioning also the lands of Coltrim. (So it looks as though William also had land in Coltrim).
PRONI’s index of pre 1858 wills lists wills for Andrew Davidson of Drumrot, probate granted 16.6.1718 & George Davidson 21.10.1727. It looks as though transcripts of both have survived and are in PRONI ref: T/748 page 100. (They are not on-line so you would either need to get someone to go to PRONI to look them up or pay PRONI to copy them for you).
As you probably already know, Moneymore 1st’s records start in 1827. Getting back earlier than that will be challenging. However the probate records from the early 1700s tell you that the families have been there since then, so they probably arrived from Scotland in the 1600s. Have you checked the Registry of Deeds records (available on Familysearch)? Your families were evidently farming in Drumrot in the 1700s and may have taken out 3 lives leases, which were popular in that century. Worth checking for as you often get ages and relationship between the 3 lives, as well as references back to previous leases and the tenant(s) then. Might help.