Hello Nannette,
Before you start taking wild guesses as to how the name might have been spelt whilst the family lived in Ireland it would be worth while to slow down and consider what you know about the family that might help in finding some links to them in Ireland ... the family's denomination would be a good starting point. Have a look at the Christian names in the family ... do any of them look like surnames?
I realise that there are many different spelling variations but I'd be inclined to stick to researching McCune before you consider other variations. Have a look at the Surname Search on the Irish Ancestors site.
www.ireland.com/ancestor/surnameI googled
"McCune Quakers" and spotted an article headed
"Vol. 8, pg 170, Ohio History" from
"The Scholarly Journal of the Ohio Historical Society" which mentions a Colonel McCune whose ancestry could be traced back to Scotland. The spelling was originally McEuen. The Valuations show that there were several families named McEwen in Ireland.
The Belfast Weekly News Extracts 1890- 1896 site records the death of W. J. T. McCune on March 16th 1890.
http://tinyurl.com/3tbntlBelfast: McCune listings in street directories ...The 1852 Belfast and Province of Ulster Directory Alexander, ship builder, 10, Pilot Street
E., milliner, 17, Academy Street
Isaac, in Mr. Gelston's spirit store, North Street, 3, Ingram Place
John, engineer, Falls Road
Mrs., 12, Queen Street
Richard, spirit dealer, 17, Albert Square
William, builder, Castle Street
The 1843 Belfast and Province of Ulster Directory David, 41, Talbot Street.
Samuel, baker, 58, Nelson Street.
Hugh, labourer, 51, Green Street.
McCune & Co., builders & plumbers, 3, Fountain Street & 12, Castle Street.
Mrs., 12, Queen Street.
Alexander, millwright, Shankhill Road.
Miss, 21, Fountain Street.
Isaac, writing clerk, 5, James Street South.
John, engineer, Pilot Street.
The 1819 Belfast and Province of Ulster Directory Hugh, bookkeeper, 95, High Street
Mrs., 12, Mill Street
Samuel, ship bread baker, 15, Mary Street
The directories are on the Lennon Wylie website.
Christopher