K,
read your post.
Did a wee bit of digging and whilst not wanting to lead anyone up a blind alley, these are my thoughts..
I think the family are east Belfast, Newtownards Road area...my reasoning is based on my own maternal grandparents who lived in the same area, at the same time.
The bulk of the Glentoran players, at that time, were local.
Boxing was, and still is, a popular sport at the Oval (the Glen's home ground).
Ledley Hall is the nearest boxing club and has strong links in the area, and, I believe is at least 100 years old.
'Davison' is commonly misspelled 'Davidson' and vice versa here..along with many names..
This area of Belfast was a major residential area for the dock, the shipyard and associated engineering works since the late 1890's.
Therefore, I can find two matches on the 1901 and 1911 censuses which possibly will help your search..
1901 -
http://www.census.nationalarchives.ie/pages/1901/Down/Victoria/Wolff_Street/1219805/1911 -
http://www.census.nationalarchives.ie/pages/1911/Down/Victoria__part_of_/Wolff_Street/227062/Note the surname change. I think William's mother may have been called 'Margaret Jane', but recorded as 'Jane' on the census returns.
Will do some further re burials etc and get back to you...
Hope this helps at this stage..
David