Possibles, for your consideration:
I started with looking in the 1851 census, nearest to when he was born.
Searched for William Aitch*, born 1845 +/- 2 years in Scotland, looking anywhere in the UK
FindMyPast(England & Scotland) and SP (Scotland only, just a free search), with another person in the house called John, as both sites present the results in a format where you can highlight the results & copy and paste them into excel, which I like as its easier to sort and add notes)
Stuck all the results into Excel and checked the transcripts on FindMyPast and Ancestry to check and compare.
Discounting any where the Dad wasn't called John I was left with 3 - one of those was in Liverpool and Dad was a printer - discounted that one. Re-checked without the other person as John in case Dad died and Mam was a widow by 1851. Discounted those results as no widow Aitch* with a child called William.
- NB this would not preclude a widow Aitch* remarrying before 1851 and the child going by his step-dad's name but that would take more firkling than I currently have time for and we have to start somewhere :-)
I was left with two possibles:
1.
1851 Dunbar, Haddingtonshire
John AITCHISON, Head, age 41, ag lab, Born Hutton, Berwickshire
Mary, wife, age 40, Born Spott, Haddingtonshire
Margaret, daughter, age 14, Ag lab, born Haddington
Jane, daughter, 12, born Haddington
Fanny, 10, age 8, born Edinburgh
Janet, daughter, age 8, born Edinburgh
William, son, age 5, born Dunbar
James, son, age 1, born Dunbar
2.
1851 Grammar Hall, 19, Coldingham, Berwickshire
John AITCHISON, head, age 47, ag lab born Coldingham
Agnes, wife, age 42, born Coldingham
Archibald, son, age 12, born, Oldhamstocks, Haddingtonshire
John, son, 11, born Coldingham
William, age 8, born Coldingham
NB if you use Ancestry, they have that second family transcribed as 'Sitskson' (go figure that one!)
SP have baptisms for:
William with parents John Aitchison and Mary (nee Muir) 09/08/1843, Dunbar
and
William with parents John Aitchison and Agnes (nee Strachan) 12/01/1842, Coldingham
Boo