"....the Irish National Quidditch Team! Presenting - Connolly! Ryan! Troy! Mullet! Moran! Quigley! Aaaaaand - Lynch!"
-- Ludo Bagman introducing the Irish team onto the pitch (GF8)
Along with fellow Beater Quigley, Connolly was part of the Irish team in the 1994 final final, all of whom were riding Firebolt brooms. He and Moran, one of the Chasers, and supported the injured Aidan Lynch when the Cup was presented to the winning Irish team members, following Bulgarian Seeker Viktor Krum’s surprise capture of the Snitch (GF8).
Connolly is an Anglicised form of the Old Gaelic name O'Conghaile. The Gaelic prefix O means "male descendant of". The name Conghaile comes from con "a hound" and gal "valour". Therefore the name could be rendered as "descendants of the Hound of Valour" (Internet Surname Database).
Like the other members of the Irish National Team, Connolly is named for one of J.K. Rowling's friends:
"The Irish Quidditch team players are all named after people I have known." (JKR)
Lexicon list of Beaters
From the Web
Writing by J.K. Rowling on Pottermore: History of the Quidditch World Cup
Pottermore feature: Quidditch collection
Harry Potter Wiki: Connolly