Green’s Dictionary of Slang

(Irish) a pet name for a child.

1914		Joyce ‘An Encounter’ Dubliners (1956) 20: Mahony used slang freely, and spoke of Father Butler as Old Bunser.
1975		E. Brady All in! All in! 3: A little girl can bask in the security of being her Daddy’s little hen [...] or his little bunser.