Green’s Dictionary of Slang

Billy Harran’s dog n.

also Billy Harren’s dog

(Irish) a time-server, one who befriends whoever they happen to be with.

[Ire]S. MacManus Rocky Road to Dublin n.p.: Every time she found him too ready to fall in with the ways of each latest advisor, it was ‘Hagh! Billy Harren’s wee dog, who went a bit of the way with everyone’ [BS].
[Ire]B. Share Naming Names 125: ‘Still like Dolan’s ass, I went a bit of the way with everybody.’ Similarly, Billy Harran’s dog.