:To Paul R. Smith
From Marcas O Murchu
Mc Weeny is an anglicization of the older Irish name Mac Suibhne or Mac Shuibhne ie descendants of Suibhne
There is a person mentioned in a Irish folklore tale called Suibhne.
It is also found as a variation Mc Sweeney (Fanaid and Lough Swilly , Donegal ) and Mahwhinney ( N Ireland and Scotland)
You did not say whether south or north Leitrim?. The island in Galway is as far as I know unconnected.However if the family say there is a connection there may possibly be a link. Have you looked at Griffith's Valuations for the 1850s or the Tithe Applotments for 1830s? Available from most records offices and county libraries.
The closest to Leitrim either the Donegal branch(north) and (south) another branch in the Strokestown area of Roscommon who, I believe, may have a coach company. Many northern surnames went south and west in the seventeenth century, but yours could, of course, have gone there for other reasons, at any date.
Good luck with the programme. My own family was from lovely Leitrim Rossinver (Meehan, Murphy , Connolly) and Arigna (Mc Partland, Daly )and just over the Sligo/Roscommon border my grandfather was a Mc Dermottroe(Kilbryan parish at Lough Key, Boyle) and also some Conlon blood thrown in for good measure (Conway's Cross/Carnagultia, Geevagh) and finally also the Daly family of Gleann, Killamey)are connected.
Regards Marcas O Murchu