You may be able to find baptism records for the children of Michael and Winnifred but not for themselves. Sligo's earliest records for CofI is 1762 1 parish of 25, next closest is 1802. For RC the earliest is 1803.
There were 25 Benson's in the Tithe Applotments of 1823-1837:
14 Rooney's in 3 Parish's Ahamlish, Ballysumaghan and Calry
There are 6 Benson's in the Parish of Ballysumaghan in 1824
names Luke, Paul, Denis, Thady, Owen.
Griffith Valuations: www.askaboutireland.ie
Denis Rourke, Parish of Ahamlish, Townland Kilkilloge
Union of Sligo (within 12 mi) tenant of Viscount Palmerson
42 a house and land 2 acres 2 roods and 34 perches
It is likely that the parents or their parents came originally from Scotland in the 'plantation of Scots farmers' to Ireland.
There are 14 Benson's in the current Sligo telephone book.
and about 70 Rooney's. Until you have some idea of the parish or place, it wouldn't be worth writing any letters.www.familysearch.org
- has records for Scotland