My Dad's still singing away at 77 - and making CDs if you don't mind!
His favourite when we were little was
'Far Away in Australia'.
Some of his favourites are:

Down By Lough Arrow's Side
Glorious Old Round Towers of Ireland
Lakes of Sligo
Three Leaf Shamrock.

He has a lot of songs - far more now than I remember from my childhood.

Cheers,

Quokka