Chris:
Civil registration of births started in 1864. Prior to 1864 we have to rely on available church records. I searched the subscription site Roots Ireland and located two possible leads for your John Cuff.
James Cuff baptized January 12 1840 father Dominic Cuff mother Catherine Raftery RC parish Fairymount/Tibohine
James Cuff baptized July 9 1840 father John Cuff mother Catherine Callahan RC parish Fairymount/Tibohine
There is another RC parish in the general area called Frenchpark but records for that RC parish do not begin until 1865.
I looked at the 1855 Griffiths Valuation head of household listing and there were four Cuffe record. There were two Johns one in Tibohine and the other in Rathkeery townland. There was no Dominic. https://www.failteromhat.com/griffiths/roscommon/tibohine.htm
I assume that James Cuff emigrated. Do you know the name of his oldest son?
Roger McDonnell