Conflicting claims of authorship have been made in this thread:

The Clancys state, in the liner notes to 'The Rising of the Moon', but with no source given for that info, that the song was written by "Terrence Ward of the Irish Times".

Fergus tells us it was written by his great-uncle Peter Ellis, according to information handed down form his grandfather via his father.

Martin Ryan's recollection is that it originated in Glasgow anonymously, again with no source given but with internet evidence for widespread agreement on this.

Are they all talking about the same song, namely the one starting "In Mountjoy gaol one Monday morning"? If so, how can these claims be reconciled? Any suggestions?

The Clancys do mean this song. Their lyrics are shorter than all of those quoted above, for what reasons I don't know:

KEVIN BARRY
(Terrence Ward / Trad)

                Chorus:
                    Another martyr for old Erin
                    Another murder for the crown
                    The British laws may crush the Irish
                    But cannot keep their spirits down

In Mountjoy gaol one Monday morning
High upon the gallows tree
Kevin Barry gave his young life
For the cause of liberty
But a lad of eighteen summers
Yet no true man can deny
As he walked to death that morning
He proudly held his head on high

Just before he faced the hangman
In his dreary prison cell
The British soldiers tortured Barry
Just because he would not tell
The names of all his brave companions
And other things they wished to know
Turn informer or we'll kill you
Kevin Barry answered, No

(Clancy Bros & Tommy Makem, The Rising of the Moon, 1956, to the - slightly modified - tune of the sea song 'Rolling Home')